3746:
2701:
2725:
3761:
3777:
3385:
8750:
8399:
1051:. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to
3741:{\displaystyle {\begin{aligned}f'(x)&=\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left(2x+dx\right)\\&=2x.\end{aligned}}}
11231:
10856:
12746:
8745:{\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos \left(x^{2}\right)-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos \left(x^{2}\right)-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}
13506:
11946:
6312:
11029:
6330:
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using
10628:
12613:
6193:
2507:
12145:
12464:
4260:
has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the
10489:
9897:
7127:
5097:
5099:
Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a
3287:
8075:
12608:
11833:
1525:
13402:
7327:
7213:
7663:
5271:
2809:
9692:
8394:
6812:
7438:
8281:
6200:
1277:
15532:
2328:
145:
6701:
8404:
3390:
12041:
11758:
4878:
15202:
13206:
13018:
12871:
9592:
10222:
10185:
6614:
7913:
6989:
6920:
1820:
6467:
13345:
11431:
7821:
4758:
14782:
10400:
6543:
7979:
14463:
12383:
11226:{\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right),}
14243:
6085:
12500:
10917:
10544:
10333:
9798:
4929:
14439:
14415:
11373:
11022:
10623:
14138:
One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all
9135:
5509:
3334:
10851:{\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}
9970:
4592:), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the
5547:
13887:
13789:
13760:
13634:
12933:
12902:
12830:
12799:
12238:
12207:
11500:
9754:
9725:
9506:
3051:
9785:
10150:
1380:
14683:
10041:
9104:
9077:
8921:
7581:
5482:
3129:
2688:
13731:
13709:
13605:
13581:
13557:
13397:
13250:
13163:
13137:
13091:
13044:
12768:
12741:{\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.}
12542:
12376:
12352:
12036:
11992:
11828:
11806:
9919:
9474:
8191:
7362:
7248:
4545:
2933:
1302:
10120:
6078:
5780:
3077:
to an infinitesimal change in its input. In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required. The system of
15407:
5749:
5354:
3380:
2323:
2214:
2180:
1720:
14267:
5922:
5185:
13921:
11321:
6844:
5979:
4308:
3139:. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the
2880:
1082:
that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the
9283:
9046:
9019:
8844:
7933:
7723:
5617:
5582:
5431:
3191:
2533:
2075:
1406:
1324:
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8876:
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6727:
6640:
6031:
5141:
4984:
4713:
1441:
16592:
10968:
10571:
10090:
9383:
9195:
8812:
8785:
5861:
14060:
13685:
5308:
3186:
1137:
11694:
6995:
4571:
4491:
4336:
4258:
4212:
4146:
4100:
14810:
14729:
14706:
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14615:
4676:
4651:
4463:
3312:
2643:
2600:
1936:
1911:
15051:
15027:
15003:
14983:. See p. 133 for the power rule, p. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, p. 338–339 for exponential with base
14836:
14108:
14086:
14029:
14009:
13989:
13963:
13941:
13848:
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13654:
13375:
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12953:
12520:
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12178:
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11469:
11295:
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11255:
10941:
10395:
10375:
10355:
10245:
10061:
9941:
9349:
9326:
9305:
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9221:
9159:
8992:
8972:
8941:
8115:
8095:
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5827:
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5696:
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4989:
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4802:
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4120:
4074:
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4034:
4014:
3994:
3972:
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3862:
3840:
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3157:
3075:
2997:
2977:
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2837:
2620:
2575:
2553:
2281:
2258:
2236:
2146:
2126:
2104:
2050:
2030:
2010:
1986:
1958:
1884:
1862:
1842:
1768:
1744:
1684:
1664:
1644:
1624:
1600:
1578:
1556:
1344:
1191:
1161:
1446:
14327:
14307:
13791:(in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is
9994:
of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in
17568:
16461:
5800:. However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.
17556:
4584:, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions (for example, if the function is a
1028:, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called
13521:
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a
4604:
in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.
12547:
7490:
are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions.
15772:
David, Claire (2018), "Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the
Weierstrass function",
1201:
15072:, p. 111–112, 119, respectively. For the special case of the product rule, that is, the product of a constant and a function, see
7254:
17:
7143:
17678:
17563:
7996:
2733:
5190:
241:
17546:
17541:
15166:
13585:. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If
1008:'s output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the
14895:
9597:
8296:
6733:
17716:
17551:
17536:
16650:
14153:
Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is
7368:
16838:
11941:{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}
17531:
6307:{\displaystyle \textstyle D_{x}^{2}f(x,y)={\frac {\partial }{\partial x}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}}
5938:. To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function
13501:{\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}
4573:. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.
9756:. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of
2955:
is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the
3766:
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a
17952:
17148:
16902:
16554:
16509:
16444:
16421:
16350:
16326:
16306:
16278:
16252:
16212:
16192:
16110:
16081:
16063:
16035:
16015:
15981:
15953:
15924:
15896:
15857:
15821:
15763:
15700:
15665:
15511:
15483:
7597:
6335:
for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as
1074:
whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The
507:
482:
6646:
10861:
9416:. Suppose that a function represents the position of an object at the time. The first derivative of that function is the
9409:
4716:
15391:
15355:
11703:
16700:
16531:
9899:
if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of
9511:
4813:
2817:
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function
1067:
983:
546:
14481:
14457:
14433:
14341:
14321:
10190:
10155:
6549:
5935:
64:
17646:
17505:
16478:
16396:
16378:
16234:
15878:
15718:
15604:
14210:
13516:
9987:
6926:
502:
220:
31:
6860:
5513:, respectively. For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in
1777:
17768:
17060:
16976:
16150:
16120:
6408:
487:
15226:
14847:
13279:
13170:
12982:
12835:
11378:
17947:
17848:
17775:
17641:
17573:
17198:
17053:
17021:
16780:
14285:
14261:
9792:
7134:
4724:
1532:
823:
497:
472:
154:
15123:
15107:
8200:
6493:
6188:{\textstyle D_{xy}f(x,y)={\frac {\partial }{\partial y}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}}
17942:
17937:
17758:
17274:
17251:
16966:
15564:
12332:. The total derivative gives a complete picture by considering all directions at once. That is, for any vector
15186:
2502:{\displaystyle {\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.}
17763:
17709:
17364:
17302:
17097:
16971:
16643:
16581:
16405:
15315:
15282:
605:
552:
433:
16174:
16850:
16828:
15427:
15266:
14841:
14736:
13796:
12140:{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}
10867:
10494:
10268:
7870:
7588:
4613:
259:
231:
17673:
16005:
11326:
10975:
10576:
7938:
4883:
2730:
The derivative at different points of a differentiable function. In this case, the derivative is equal to
342:
17658:
17424:
17038:
16860:
16576:
3319:
1108:
856:
464:
302:
274:
38:
17043:
16813:
14147:
13021:
5520:
4627:
1044:
727:
691:
468:
347:
236:
226:
13863:
13765:
13736:
13610:
12909:
12878:
12806:
12775:
12469:
12214:
12183:
11476:
9730:
9701:
9482:
17462:
17409:
9759:
6388:
5514:
3010:
1040:
491:
16870:
10125:
1349:
327:
17702:
17578:
17349:
16897:
16636:
12502:
is chosen so that the error in this approximation is as small as possible. The total derivative of
10484:{\displaystyle {\frac {\partial f}{\partial x}}=2x+y,\qquad {\frac {\partial f}{\partial y}}=x+2y.}
10005:
9948:
8881:
7764:
3096:
2128:
sometimes has a derivative at most, but not all, points of its domain. The function whose value at
626:
186:
14651:
13714:
13692:
13588:
13564:
13540:
13380:
13233:
13146:
13120:
13074:
13027:
12751:
12525:
12359:
12335:
12019:
11975:
11811:
11789:
9902:
9457:
8143:
7332:
7218:
4500:
3135:
are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called
2885:
1287:
17344:
17016:
16616:
15176:
12459:{\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}
9401:
function is infinitely differentiable; taking derivatives repeatedly will eventually result in a
9354:
6851:
6038:
5756:
1140:
1024:
of the function near that input value. For this reason, the derivative is often described as the
940:
732:
621:
16571:
15220:
5725:
3789:
since the tangent slopes do not approach the same value from the left as they do from the right.
3343:
2286:
1035:
There are multiple different notations for differentiation, two of the most commonly used being
17880:
17780:
17472:
17354:
17175:
17123:
16929:
16907:
16775:
15079:
14943:
11763:
11446:
11298:
10002:. As with ordinary derivatives, multiple notations exist: the partial derivative of a function
9452:
9441:
5888:
5146:
3337:
3132:
1005:
976:
905:
866:
750:
686:
610:
15655:
15421:
15385:
15381:
15365:
15325:
15297:
15212:
15133:
14905:
14879:
14857:
13894:
11303:
9892:{\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}
7546:
6817:
5943:
4268:
2844:
2650:
17873:
17868:
17832:
17828:
17753:
17726:
17598:
17457:
17369:
17026:
16961:
16934:
16924:
16845:
16833:
16818:
16790:
16523:
15417:
15401:
15276:
15236:
15117:
14787:
14220:
14184:
14113:
10227:
among other possibilities. It can be thought of as the rate of change of the function in the
9999:
9255:
8817:
7918:
7708:
6325:
5797:
5789:
5589:
5554:
5317:
4986:. These are abbreviations for multiple applications of the derivative operator; for example,
4807:
3340:, which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function
2512:
1385:
1309:
1071:
950:
616:
387:
332:
293:
199:
15437:
14132:
12287:
10095:
8849:
7730:
7122:{\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x)}
6706:
6619:
6006:
5111:
4954:
4683:
1411:
17900:
17805:
17414:
17033:
16880:
16495:
15941:
15068:, p. 161, 164, respectively. For the product rule, quotient rule, and chain rule, see
14205:
14176:. The study of differential calculus is unified with the calculus of finite differences in
14154:
13800:
13522:
13350:
10946:
10920:
10549:
10068:
9361:
9168:
8790:
8763:
8135:
6475:
6082:. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g.
5834:
5671:
5325:
5101:
5092:{\textstyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}{\Bigl (}{\frac {d}{dx}}f(x){\Bigr )}.}
4593:
3136:
2185:
2151:
1989:
1691:
1164:
1087:
1079:
1021:
955:
935:
861:
530:
449:
423:
337:
16370:
The Words of
Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
15587:
14036:
13661:
10263:, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee". For example, let
9115:
6347:
The following are the rules for the derivatives of the most common basic functions. Here,
5784:, respectively. This notation is used exclusively for derivatives with respect to time or
5489:
5284:
3162:
1113:
8:
17895:
17885:
17838:
17434:
17359:
17246:
17203:
16954:
16939:
16770:
16758:
16745:
16705:
16685:
16540:
16146:
14813:
14215:
14192:
14177:
14166:
14162:
14139:
11673:
9084:
9057:
5462:
4581:
4550:
4470:
4315:
4237:
4191:
4125:
4079:
3865:
2707:
1196:
1017:
930:
900:
890:
777:
631:
428:
284:
167:
162:
17694:
15945:
15196:
14792:
14711:
14688:
14628:
14597:
9026:
8999:
5411:
4658:
4633:
4445:
3294:
2625:
2582:
2055:
1918:
1893:
17917:
17800:
17523:
17498:
17329:
17282:
17223:
17188:
17183:
17163:
17158:
17153:
17118:
17065:
17048:
16949:
16823:
16808:
16753:
16720:
16500:
16434:
16226:
15827:
15781:
15643:
15517:
15036:
15012:
14988:
14821:
14158:
14128:
14093:
14071:
14014:
13994:
13974:
13948:
13926:
13833:
13813:
13639:
13360:
13257:
13211:
13098:
13049:
12960:
12938:
12505:
12315:
12265:
12245:
12163:
11999:
11953:
11769:
11651:
11631:
11611:
11591:
11569:
11547:
11527:
11507:
11454:
11280:
11260:
11240:
10926:
10380:
10360:
10340:
10230:
10046:
9991:
9981:
9926:
9788:
9334:
9311:
9290:
9233:
9206:
9144:
8977:
8957:
8926:
8100:
8080:
7848:
7826:
7688:
7668:
7522:
7500:
7473:
7453:
6370:
6350:
5986:
5866:
5812:
5703:
5681:
5648:
5624:
5440:
5387:
5363:
4934:
4787:
4765:
4589:
4423:
4403:
4383:
4363:
4343:
4217:
4171:
4151:
4105:
4059:
4039:
4019:
3999:
3979:
3957:
3935:
3911:
3891:
3871:
3847:
3825:
3799:
3767:
3142:
3060:
2982:
2962:
2938:
2822:
2605:
2560:
2538:
2266:
2243:
2221:
2131:
2111:
2089:
2035:
2015:
1995:
1971:
1943:
1869:
1847:
1827:
1753:
1729:
1669:
1649:
1629:
1609:
1585:
1563:
1541:
1329:
1176:
1146:
1083:
895:
798:
782:
722:
717:
712:
676:
557:
476:
382:
377:
181:
176:
14146:. The idea is to embed the continuous functions in a larger space called the space of
14142:
functions and many other functions can be differentiated using a concept known as the
17890:
17853:
17663:
17487:
17419:
17241:
17218:
17092:
17085:
16988:
16803:
16695:
16599:
16550:
16527:
16505:
16474:
16440:
16417:
16392:
16374:
16346:
16322:
16302:
16274:
16248:
16230:
16208:
16188:
16106:
16077:
16059:
16031:
16011:
15977:
15949:
15920:
15892:
15874:
15853:
15831:
15817:
15759:
15724:
15714:
15696:
15661:
15600:
15521:
15507:
15479:
14594:
In the formulation of calculus in terms of limits, various authors have assigned the
14173:
9402:
9049:
6479:
4623:
4585:
4577:
969:
803:
581:
459:
412:
269:
264:
14863:
14578:
14576:
14574:
17843:
17823:
17621:
17404:
17317:
17297:
17228:
17138:
17080:
17072:
17006:
16919:
16680:
16675:
16453:
16294:
16266:
16165:
16135:
16098:
16051:
15969:
15912:
15845:
15809:
15791:
15751:
15688:
15638:
15633:
15625:
15583:
15573:
15544:
15499:
13530:
12155:
9695:
9107:
4619:
4494:
3078:
2718:
of the tangent line is equal to the derivative of the function at the marked point.
1059:, how the position changes as time advances, the second derivative is the object's
1036:
813:
707:
681:
542:
454:
418:
14157:. Here, it consists of the derivation of some topics in abstract algebra, such as
12312:, no single directional derivative can give a complete picture of the behavior of
17962:
17858:
17795:
17790:
17785:
17683:
17668:
17452:
17307:
17287:
17256:
17233:
17213:
17107:
16763:
16710:
16544:
16368:
16336:
16316:
16288:
16202:
16045:
16025:
15991:
15963:
15906:
15839:
15747:
15735:
15493:
14571:
14143:
13807:
13354:
9995:
9437:
4493:. Even a function with a smooth graph is not differentiable at a point where its
3086:
1091:
1075:
945:
818:
772:
767:
654:
567:
512:
16602:
13349:, then the total derivative can be expressed using the partial derivatives as a
9420:
of an object with respect to time, the second derivative of the function is the
3282:{\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}
17957:
17593:
17492:
17339:
17292:
17193:
16996:
16430:
16410:
16290:
Fractional and
Multivariable Calculus: Model Building and Optimization Problems
15866:
15613:
15467:
14230:
14065:
13855:
13534:
11524:
measure its variation in the direction of the coordinate axes. For example, if
9425:
5931:
5276:
4262:
3817:
1528:
828:
636:
403:
17011:
16298:
16270:
16182:
16169:
16139:
16102:
16055:
15973:
15916:
15849:
15813:
15755:
15692:
15503:
15474:
Calculus, Vol. 1: One-Variable
Calculus with an Introduction to Linear Algebra
9795:, whose coordinates are the derivatives of the coordinate functions. That is,
17931:
17862:
17818:
17467:
17322:
17208:
16912:
16887:
16620:
15796:
15728:
15528:
15472:
13851:
7987:
4597:
3929:
3054:
2724:
1168:
808:
572:
322:
279:
3782:
The absolute value function is continuous but fails to be differentiable at
17477:
17447:
17312:
16875:
16588:
16090:
16001:
15965:
Fractional
Dynamic Calculus and Fractional Dynamic Equations on Time Scales
15578:
15559:
15549:
14124:
14120:
12603:{\displaystyle f'(\mathbf {a} )\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}
12149:
11434:
9477:
9421:
8757:
7755:
4600:
proved that the set of functions that have a derivative at some point is a
2711:
2240:. It is still a function, but its domain may be smaller than the domain of
1060:
562:
307:
16243:
Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006),
5459:
prime". Similarly, the second and the third derivatives can be written as
16725:
16667:
16470:
14399:
9508:. A vector-valued function can be split up into its coordinate functions
3090:
1280:
997:
925:
14359:
11297:
is differentiable at every point in some domain, then the gradient is a
4576:
Most functions that occur in practice have derivatives at all points or
4467:. This can be seen graphically as a "kink" or a "cusp" in the graph at
2645:. So, the derivative of the squaring function is the doubling function:
1520:{\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,}
17442:
17374:
17128:
17001:
16865:
16855:
15739:
15647:
15498:, Springer Optimization and Its Applications, vol. 164, Springer,
13968:
9398:
9393:
8753:
8130:
7322:{\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}}
6399:
5785:
5105:
4601:
2216:
is defined and elsewhere is undefined is also called the derivative of
1048:
671:
595:
317:
312:
14487:
14291:
13795:
and this imposes relations between the partial derivatives called the
7208:{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}
4806:". This derivative can alternately be treated as the application of a
17636:
17384:
17379:
16690:
16607:
16504:(Revised, Updated, Expanded ed.), New York: St. Martin's Press,
16457:
16342:
15933:
8070:{\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}
6483:
5266:{\textstyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}
3131:
for any finite number of terms. Such numbers are infinite, and their
2804:{\displaystyle \sin \left(x^{2}\right)+2x^{2}\cos \left(x^{2}\right)}
2700:
1004:
is a fundamental tool that quantifies the sensitivity of change of a
600:
590:
16027:
Modern
Differential Geometry of Curves and Surfaces with Mathematica
15629:
8954:
are the result of differentiating a function repeatedly. Given that
17905:
17743:
17738:
17631:
17133:
16659:
15786:
15254:
14225:
11698:. These are measured using directional derivatives. Given a vector
11234:
9424:
of an object with respect to time, and the third derivative is the
9417:
3760:
3082:
1056:
666:
408:
365:
54:
13095:, then all the partial derivatives and directional derivatives of
11648:
direction. However, they do not directly measure the variation of
17482:
16735:
14188:
9413:
5793:
5678:
a dot is placed over a symbol to represent a time derivative. If
3776:
2956:
1646:. Multiple notations for the derivative exist. The derivative of
1013:
14915:
14150:
and only require that a function is differentiable "on average".
9687:{\displaystyle \mathbf {y} =(y_{1}(t),y_{2}(t),\dots ,y_{n}(t))}
9353:
th derivative is continuous, then the function is said to be of
8389:{\displaystyle f(x)=x^{4}+\sin \left(x^{2}\right)-\ln(x)e^{x}+7}
6807:{\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}}
17651:
16715:
16518:
Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007),
1070:. In this generalization, the derivative is reinterpreted as a
15911:, Lecture Notes in Physics Monographs, vol. 8, Springer,
14812:
is defined as an infinitesimal. It is also interpreted as the
7433:{\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}
5863:
and higher derivatives are written with a superscript, so the
2999:. In other words, the derivative is the slope of the tangent.
16730:
13891:. The derivative (or differential) of a (differentiable) map
10248:
2715:
1009:
9405:, and all subsequent derivatives of that function are zero.
9387:. A function that has infinitely many derivatives is called
5275:
Another common notation for differentiation is by using the
16628:
16361:Über eine Funktion aus Bolzanos handschriftlichem Nachlasse
13537:, such as functions from (a domain in) the complex numbers
12803:, so the norm in the denominator is the standard length on
5807:, which represents the differential operator by the symbol
4626:
in 1675, which denotes a derivative as the quotient of two
2325:. Then the quotient in the definition of the derivative is
1052:
16265:, Graduate Texts in Mathematics, vol. 218, Springer,
14559:
12906:, and the norm in the numerator is the standard length on
12262:
in a chosen direction is the best linear approximation to
5722:, then the first and second derivatives can be written as
5517:, whereas others place the number in parentheses, such as
4618:
One common way of writing the derivative of a function is
1988:
is a function that has a derivative at every point in its
17724:
15533:"Uber die Baire'sche Kategorie gewisser Funktionenmengen"
15156:
15154:
14964:
14962:
14387:
9476:
of a real variable sends real numbers to vectors in some
4360:
is positive, then the slope of the secant line from 0 to
4214:
is on the high part of the step, so the secant line from
1090:
of several variables, the
Jacobian matrix reduces to the
15938:
Mathematical
Methods for Optical Physics and Engineering
15055:, and p. 369 for the inverse of trigonometric functions.
14375:
12150:
Total derivative, total differential and
Jacobian matrix
11668:
in any other direction, such as along the diagonal line
11588:, then its partial derivatives measure the variation in
5586:. The latter notation generalizes to yield the notation
4102:
is on the low part of the step, so the secant line from
1992:, then a function can be defined by mapping every point
1272:{\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
16318:
A Mathematical Course for
Political and Social Research
15343:
13529:
An important generalization of the derivative concerns
3093:
that contain numbers greater than anything of the form
16546:
Foundations of Differentiable Manifolds and Lie Groups
16242:
15331:
15151:
15139:
14959:
14740:
14654:
14535:
14347:
7658:{\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'}
6204:
6088:
5193:
4992:
4886:
4816:
4728:
3013:
1781:
16184:
Elementary Calculus: An Approach Using Infinitesimals
15683:
Choudary, A. D. R.; Niculescu, Constantin P. (2014),
15289:
15287:
15285:
15242:
15039:
15015:
14991:
14824:
14795:
14739:
14714:
14691:
14631:
14600:
14547:
14119:
Differentiation can also be defined for maps between
14096:
14074:
14039:
14017:
13997:
13977:
13951:
13929:
13897:
13866:
13836:
13816:
13768:
13739:
13717:
13695:
13664:
13642:
13613:
13591:
13567:
13543:
13405:
13383:
13363:
13282:
13260:
13236:
13214:
13173:
13149:
13123:
13101:
13077:
13052:
13030:
12985:
12963:
12941:
12912:
12881:
12838:
12809:
12778:
12754:
12616:
12550:
12528:
12508:
12472:
12386:
12362:
12338:
12318:
12290:
12268:
12248:
12217:
12186:
12166:
12044:
12022:
12002:
11978:
11956:
11836:
11814:
11792:
11772:
11706:
11676:
11654:
11634:
11614:
11594:
11572:
11550:
11530:
11510:
11479:
11457:
11381:
11329:
11306:
11283:
11263:
11243:
11032:
10978:
10949:
10929:
10870:
10631:
10579:
10552:
10497:
10403:
10383:
10363:
10343:
10271:
10233:
10193:
10158:
10128:
10098:
10071:
10049:
10008:
9951:
9929:
9905:
9801:
9762:
9733:
9704:
9600:
9514:
9485:
9460:
9364:
9337:
9314:
9293:
9258:
9236:
9209:
9171:
9147:
9118:
9087:
9060:
9029:
9002:
8980:
8960:
8929:
8884:
8852:
8820:
8793:
8766:
8402:
8299:
8203:
8146:
8103:
8083:
7999:
7941:
7921:
7873:
7851:
7829:
7767:
7733:
7711:
7691:
7671:
7600:
7549:
7525:
7503:
7476:
7456:
7371:
7335:
7257:
7221:
7146:
6998:
6929:
6863:
6820:
6736:
6709:
6649:
6622:
6552:
6496:
6411:
6373:
6353:
6203:
6041:
6009:
5989:
5946:
5891:
5869:
5837:
5815:
5759:
5728:
5706:
5684:
5651:
5627:
5592:
5557:
5523:
5492:
5465:
5443:
5414:
5390:
5366:
5328:
5287:
5149:
5114:
4957:
4937:
4790:
4768:
4727:
4686:
4661:
4636:
4553:
4503:
4473:
4448:
4426:
4406:
4386:
4366:
4346:
4318:
4271:
4240:
4220:
4194:
4174:
4154:
4128:
4108:
4082:
4062:
4042:
4022:
4002:
3982:
3960:
3938:
3914:
3894:
3874:
3850:
3828:
3802:
3388:
3346:
3322:
3297:
3194:
3165:
3145:
3099:
3085:
and infinitesimal quantities. The hyperreals are an
3063:
2985:
2965:
2941:
2888:
2847:
2825:
2736:
2653:
2628:
2608:
2585:
2563:
2541:
2515:
2331:
2289:
2269:
2246:
2224:
2188:
2154:
2134:
2114:
2092:
2058:
2038:
2018:
1998:
1974:
1946:
1921:
1896:
1872:
1850:
1830:
1780:
1756:
1732:
1694:
1672:
1652:
1632:
1612:
1588:
1566:
1544:
1449:
1414:
1388:
1352:
1332:
1312:
1290:
1204:
1179:
1149:
1116:
67:
16517:
16047:
Derivatives and Integrals of Multivariable Functions
15616:(1923), "The History of Notations of the Calculus",
15260:
15089:
15073:
15069:
14980:
14953:
14885:
14618:
14582:
14064:. The derivative function becomes a map between the
13830:
is a space that can be approximated near each point
11996:, then they determine the directional derivative of
8760:. The known derivatives of the elementary functions
6696:{\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}}
4880:
Higher derivatives are expressed using the notation
4400:
is negative, then the slope of the secant line from
16597:
16024:Gray, Alfred; Abbena, Elsa; Salamon, Simon (2006),
15443:
15095:
12282:at that point and in that direction. However, when
5934:did not use it, and the notation was introduced by
3750:
16409:
15908:Polynomial Approximation of Differential Equations
15471:
15045:
15021:
14997:
14931:
14830:
14804:
14776:
14723:
14700:
14677:
14640:
14609:
14102:
14080:
14054:
14023:
14003:
13983:
13957:
13935:
13915:
13881:
13842:
13822:
13806:Another generalization concerns functions between
13783:
13754:
13725:
13703:
13679:
13648:
13628:
13599:
13575:
13551:
13500:
13391:
13369:
13339:
13266:
13244:
13220:
13200:
13157:
13131:
13107:
13085:
13058:
13038:
13012:
12969:
12947:
12927:
12896:
12865:
12824:
12793:
12762:
12740:
12602:
12536:
12514:
12494:
12458:
12370:
12346:
12324:
12302:
12274:
12254:
12232:
12201:
12172:
12139:
12030:
12008:
11986:
11962:
11940:
11822:
11800:
11778:
11753:{\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}
11752:
11688:
11660:
11640:
11620:
11600:
11578:
11556:
11536:
11516:
11494:
11463:
11425:
11367:
11315:
11289:
11269:
11249:
11225:
11016:
10962:
10935:
10911:
10850:
10617:
10565:
10538:
10483:
10389:
10369:
10349:
10327:
10239:
10216:
10179:
10144:
10114:
10084:
10055:
10035:
9964:
9935:
9913:
9891:
9779:
9748:
9719:
9686:
9586:
9500:
9468:
9377:
9343:
9320:
9299:
9277:
9242:
9215:
9189:
9153:
9129:
9098:
9071:
9040:
9013:
8986:
8966:
8935:
8915:
8870:
8838:
8806:
8779:
8744:
8388:
8275:
8185:
8109:
8089:
8069:
7973:
7927:
7907:
7857:
7835:
7815:
7739:
7717:
7697:
7677:
7657:
7575:
7531:
7509:
7482:
7462:
7432:
7356:
7321:
7242:
7207:
7121:
6983:
6914:
6838:
6806:
6721:
6695:
6634:
6608:
6537:
6461:
6379:
6359:
6306:
6187:
6072:
6025:
5995:
5973:
5916:
5875:
5855:
5821:
5774:
5743:
5712:
5690:
5657:
5633:
5611:
5576:
5541:
5503:
5476:
5449:
5425:
5396:
5372:
5348:
5302:
5265:
5179:
5135:
5091:
4978:
4943:
4923:
4873:{\textstyle {\frac {dy}{dx}}={\frac {d}{dx}}f(x).}
4872:
4796:
4774:
4752:
4707:
4670:
4645:
4565:
4539:
4485:
4457:
4432:
4412:
4392:
4372:
4352:
4330:
4302:
4252:
4226:
4206:
4180:
4160:
4140:
4114:
4094:
4068:
4048:
4028:
4008:
3988:
3966:
3944:
3920:
3900:
3880:
3856:
3834:
3808:
3740:
3374:
3328:
3306:
3281:
3180:
3151:
3123:
3069:
3045:
2991:
2971:
2947:
2927:
2874:
2831:
2803:
2682:
2637:
2614:
2594:
2579:, the closer this expression becomes to the value
2569:
2547:
2527:
2509:The division in the last step is valid as long as
2501:
2317:
2275:
2252:
2230:
2208:
2174:
2140:
2120:
2098:
2069:
2044:
2024:
2004:
1980:
1952:
1930:
1905:
1878:
1856:
1836:
1814:
1762:
1738:
1714:
1678:
1658:
1638:
1618:
1594:
1572:
1550:
1519:
1435:
1400:
1374:
1338:
1318:
1296:
1271:
1185:
1155:
1131:
1086:with respect to the independent variables. For a
1039:and prime notation. Leibniz notation, named after
139:
16023:
15682:
15361:
15303:
14901:
14507:
10491:In general, the partial derivative of a function
9587:{\displaystyle y_{1}(t),y_{2}(t),\dots ,y_{n}(t)}
9230:, the generalization of derivative of a function
7867:. As a special case, this rule includes the fact
6389:the base of the natural logarithm, approximately
6298:
6258:
6180:
6140:
5081:
5047:
3159:in the Leibniz notation. Thus, the derivative of
17929:
15774:Proceedings of the International Geometry Center
12618:
11869:
10698:
10217:{\displaystyle {\frac {\partial f}{\partial x}}}
10180:{\displaystyle {\frac {\partial }{\partial x}}f}
9825:
9139:. By continuing this process, if it exists, the
8974:is a differentiable function, the derivative of
6609:{\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a)}
1212:
1047:, whereas prime notation is written by adding a
140:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
16089:
15838:Debnath, Lokenath; Shah, Firdous Ahmad (2015),
14529:
13733:is certainly differentiable as a function from
13274:is written using coordinate functions, so that
12466:Similarly with the single-variable derivative,
7445:
6984:{\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}
4715:is viewed as a functional relationship between
4168:tends to zero, the slope tends to infinity. If
16187:(2nd ed.), Prindle, Weber & Schmidt,
16072:Henle, James M.; Kleinberg, Eugene M. (2003),
16071:
14617:symbol various meanings. Some authors such as
14409:
14187:involves the function that is defined for the
14172:The discrete equivalent of differentiation is
11026:, these partial derivatives define the vector
6915:{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}
4680:. It is still commonly used when the equation
1815:{\displaystyle \textstyle {\frac {df}{dx}}(a)}
17710:
16644:
16452:
16429:
16286:
15889:An Introduction to the History of Mathematics
15744:Introduction to Calculus and Analysis, Vol. 1
15232:
14477:
14473:
14453:
14449:
14429:
14425:
14337:
14317:
14301:
14281:
14257:
6462:{\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}}
1279:exists. This means that, for every positive
1063:, how the velocity changes as time advances.
977:
15780:(2), Academy of Sciences of Ukraine: 53–68,
13340:{\displaystyle f=(f_{1},f_{2},\dots ,f_{m})}
13201:{\displaystyle f'(\mathbf {a} )\mathbf {v} }
13013:{\displaystyle f'(\mathbf {a} )\mathbf {v} }
12866:{\displaystyle f'(\mathbf {a} )\mathbf {h} }
12726:
12718:
12713:
12638:
11426:{\displaystyle \nabla f(a_{1},\dots ,a_{n})}
8752:Here the second term was computed using the
6342:
1020:at that point. The tangent line is the best
16366:
16314:
15837:
15734:
15708:
15180:
15113:
14869:
14853:
14565:
14195:. This is an analogy with the product rule.
8946:
4753:{\displaystyle \textstyle {\frac {dy}{dx}}}
3976:, and returns a different value 10 for all
17717:
17703:
16651:
16637:
16247:(4th ed.), Houghton Mifflin Company,
15806:Vectors and Functions of Several Variables
14648:by itself, but only as part of the symbol
12380:, the linear approximation formula holds:
11440:
11433:. Consequently, the gradient determines a
10337:, then the partial derivative of function
9446:
8276:{\displaystyle f'(x)=h'(g(x))\cdot g'(x).}
6538:{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}
984:
970:
27:Instantaneous rate of change (mathematics)
17679:Regiomontanus' angle maximization problem
16386:
16315:Moore, Will H.; Siegel, David A. (2013),
16204:Differential Algebra And Algebraic Groups
16158:Časopis pro Pěstování Matematiky a Fyziky
16128:Časopis pro Pěstování Matematiky a Fyziky
15841:Wavelet Transforms and Their Applications
15795:
15785:
15673:
15637:
15577:
15548:
15208:
15129:
14127:, in which those generalizations are the
13869:
13771:
13742:
13719:
13697:
13616:
13593:
13569:
13545:
12915:
12884:
12812:
12781:
12590:
12575:
12220:
12189:
11482:
10860:This is fundamental for the study of the
9736:
9707:
9488:
5983:, its partial derivative with respect to
4340:, but it is not differentiable there. If
100:
17522:
16494:
16373:, Mathematical Association of American,
16245:Calculus: Early Transcendental Functions
16220:
15961:
15597:Mathematics for Economics & Business
15594:
15433:
15216:
14925:
14381:
14365:
8293:The derivative of the function given by
2602:. The limit exists, and for every input
17027:Differentiating under the integral sign
16404:
16358:
16334:
16200:
16180:
15989:
15557:
15491:
15466:
15449:
15413:
15377:
15337:
15192:
15172:
15160:
15145:
15085:
15065:
15007:, p. 343 for the exponential with base
14968:
14949:
14909:
14873:
14708:as an independent variable, and define
14622:
14541:
14501:
14405:
14393:
14369:
14277:
14273:
14253:
14249:
13810:. Intuitively speaking such a manifold
9410:application of higher-order derivatives
9163:th derivative is the derivative of the
6319:
3002:
508:Differentiating under the integral sign
14:
17930:
16539:
16287:Mathai, A. M.; Haubold, H. J. (2017),
16145:
16043:
16000:
15904:
15803:
15653:
15612:
15527:
15397:
15321:
15293:
15272:
15101:
14937:
14889:
14553:
14525:
14497:
14469:
14445:
14421:
14353:
14333:
14313:
14297:
9975:
9431:
8288:
4497:: For instance, the function given by
1602:exists, then this limit is called the
1306:, there exists a positive real number
17698:
16903:Inverse functions and differentiation
16632:
16598:
16389:Essential Calculus: With Applications
16118:
15865:
15771:
15560:"Remarks on an arithmetic derivative"
15478:, vol. 1 (2nd ed.), Wiley,
15031:, p. 344 for the logarithm with base
14921:
14777:{\displaystyle \textstyle du=dxf'(x)}
14513:
14493:
12242:, then the directional derivative of
12180:is a function from an open subset of
10912:{\displaystyle f(x_{1},\dots ,x_{n})}
10539:{\displaystyle f(x_{1},\dots ,x_{n})}
10328:{\displaystyle f(x,y)=x^{2}+xy+y^{2}}
10259:. To distinguish it from the letter
7908:{\displaystyle (\alpha f)'=\alpha f'}
5320:. The first derivative is written as
4924:{\textstyle {\frac {d^{n}y}{dx^{n}}}}
4719:. The first derivative is denoted by
3057:change in the output of the function
1043:, is represented as the ratio of two
15932:
15886:
15349:
15248:
15064:For constant rule and sum rule, see
14625:, p. 177 do not assign a meaning to
13689:then a differentiable function from
12544:is the unique linear transformation
11368:{\displaystyle (a_{1},\dots ,a_{n})}
11017:{\displaystyle (a_{1},\dots ,a_{n})}
10618:{\displaystyle (a_{1},\dots ,a_{n})}
8994:is the first derivative, denoted as
7974:{\displaystyle \alpha 'f=0\cdot f=0}
5926:. This notation is sometimes called
16260:
15657:A History of Mathematical Notations
15599:(2nd ed.), Excel Books India,
15384:for the Gateaux derivative, and p.
15309:
10862:functions of several real variables
9972:is another vector-valued function.
4717:dependent and independent variables
3329:{\displaystyle \operatorname {st} }
3007:One way to think of the derivative
1527:where the vertical bars denote the
1068:functions of several real variables
24:
16701:Free variables and bound variables
15993:Analytic Geometry and the Calculus
15261:Varberg, Purcell & Rigdon 2007
15090:Varberg, Purcell & Rigdon 2007
15074:Varberg, Purcell & Rigdon 2007
15070:Varberg, Purcell & Rigdon 2007
14981:Varberg, Purcell & Rigdon 2007
14954:Varberg, Purcell & Rigdon 2007
14886:Varberg, Purcell & Rigdon 2007
14619:Varberg, Purcell & Rigdon 2007
14583:Varberg, Purcell & Rigdon 2007
13808:differentiable or smooth manifolds
13510:
13466:
13451:
13069:If the total derivative exists at
12118:
12110:
11950:If all the partial derivatives of
11504:, then the partial derivatives of
11382:
11307:
11164:
11156:
11093:
11085:
11033:
10643:
10635:
10454:
10446:
10415:
10407:
10205:
10197:
10165:
10161:
10130:
6269:
6265:
6247:
6243:
6151:
6147:
6129:
6125:
5831:. The first derivative is written
5533:
5530:
2012:to the value of the derivative of
1066:Derivatives can be generalized to
49:Part of a series of articles about
25:
17974:
17506:The Method of Mechanical Theorems
16593:"Newton, Leibniz, and Usain Bolt"
16564:
16097:, Springer-Verlag, Theorem 17.8,
15873:, American Mathematical Society,
13517:Generalizations of the derivative
13208:is the directional derivative of
9307:successive derivatives is called
5542:{\displaystyle f^{\mathrm {iv} }}
3932:that returns the value 1 for all
2714:to that graph, drawn in red. The
32:Derivation (differential algebra)
17061:Partial fractions in integration
16977:Stochastic differential equation
16263:Introduction to Smooth Manifolds
15887:Eves, Howard (January 2, 1990),
15660:, vol. 2, Cosimo Classics,
13882:{\displaystyle \mathbb {R} ^{3}}
13854:: the prototypical example is a
13784:{\displaystyle \mathbb {R} ^{2}}
13755:{\displaystyle \mathbb {R} ^{2}}
13629:{\displaystyle \mathbb {R} ^{2}}
13434:
13418:
13385:
13238:
13194:
13186:
13151:
13125:
13079:
13032:
13006:
12998:
12928:{\displaystyle \mathbb {R} ^{m}}
12897:{\displaystyle \mathbb {R} ^{m}}
12859:
12851:
12825:{\displaystyle \mathbb {R} ^{n}}
12794:{\displaystyle \mathbb {R} ^{n}}
12756:
12722:
12706:
12698:
12676:
12656:
12648:
12623:
12563:
12530:
12495:{\displaystyle f'(\mathbf {a} )}
12485:
12449:
12441:
12419:
12402:
12394:
12364:
12340:
12233:{\displaystyle \mathbb {R} ^{m}}
12202:{\displaystyle \mathbb {R} ^{n}}
12066:
12051:
12024:
11980:
11922:
11905:
11894:
11858:
11843:
11816:
11794:
11708:
11495:{\displaystyle \mathbb {R} ^{n}}
9954:
9907:
9867:
9844:
9804:
9764:
9749:{\displaystyle \mathbb {R} ^{3}}
9720:{\displaystyle \mathbb {R} ^{2}}
9602:
9501:{\displaystyle \mathbb {R} ^{n}}
9462:
3888:. As an example, choose a point
3775:
3759:
3751:Continuity and differentiability
3046:{\textstyle {\frac {df}{dx}}(a)}
2723:
2699:
17849:Least-squares spectral analysis
17776:Fundamental theorem of calculus
17199:Jacobian matrix and determinant
17054:Tangent half-angle substitution
17022:Fundamental theorem of calculus
15371:
15362:Gray, Abbena & Salamon 2006
15058:
14974:
14588:
14519:
13525:of the function at that point.
10442:
10357:with respect to both variables
9780:{\displaystyle \mathbf {y} (t)}
7135:Inverse trigonometric functions
5936:Louis François Antoine Arbogast
3289:for an arbitrary infinitesimal
17275:Arithmetico-geometric sequence
16967:Ordinary differential equation
16387:Silverman, Richard A. (1989),
16321:, Princeton University Press,
15940:, Cambridge University Press,
15871:Partial Differential Equations
15565:Canadian Mathematical Bulletin
14770:
14764:
14049:
14043:
13923:between manifolds, at a point
13907:
13422:
13414:
13334:
13289:
13190:
13182:
13002:
12994:
12855:
12847:
12710:
12702:
12694:
12680:
12672:
12666:
12660:
12644:
12627:
12585:
12567:
12559:
12489:
12481:
12445:
12437:
12423:
12415:
12406:
12390:
12070:
12062:
11926:
11918:
11909:
11890:
11876:
11862:
11854:
11747:
11715:
11420:
11388:
11362:
11330:
11212:
11180:
11141:
11109:
11071:
11039:
11011:
10979:
10906:
10874:
10836:
10785:
10776:
10719:
10705:
10691:
10659:
10612:
10580:
10533:
10501:
10287:
10275:
10145:{\displaystyle \partial _{x}f}
10030:
10012:
9877:
9871:
9860:
9848:
9832:
9818:
9812:
9774:
9768:
9694:. This includes, for example,
9681:
9678:
9672:
9650:
9644:
9628:
9622:
9609:
9581:
9575:
9553:
9547:
9531:
9525:
9270:
9264:
9184:
9172:
8897:
8891:
8865:
8859:
8833:
8827:
8722:
8716:
8584:
8578:
8451:
8439:
8421:
8415:
8367:
8361:
8309:
8303:
8267:
8261:
8247:
8244:
8238:
8232:
8218:
8212:
8180:
8177:
8171:
8165:
8156:
8150:
7884:
7874:
7778:
7768:
7620:
7601:
7564:
7558:
7399:
7393:
7285:
7279:
7174:
7168:
7116:
7110:
7082:
7076:
7051:
7045:
7026:
7020:
6978:
6972:
6957:
6951:
6909:
6903:
6891:
6885:
6798:
6792:
6771:
6765:
6677:
6671:
6603:
6597:
6293:
6281:
6235:
6223:
6175:
6163:
6117:
6105:
6067:
6055:
5968:
5956:
5911:
5905:
5850:
5844:
5604:
5598:
5569:
5563:
5343:
5337:
5297:
5291:
5174:
5171:
5165:
5159:
5130:
5124:
5076:
5070:
4973:
4967:
4864:
4858:
4702:
4696:
4513:
4507:
4296:
4288:
4281:
4275:
3649:
3639:
3559:
3549:
3471:
3461:
3407:
3401:
3356:
3350:
3261:
3255:
3246:
3231:
3209:
3203:
3175:
3169:
3040:
3034:
2922:
2919:
2907:
2889:
2869:
2866:
2860:
2848:
2668:
2662:
2393:
2380:
2365:
2359:
2350:
2338:
2299:
2293:
2203:
2197:
2169:
2163:
1808:
1802:
1709:
1703:
1494:
1488:
1479:
1467:
1430:
1418:
1375:{\displaystyle |h|<\delta }
1362:
1354:
1260:
1254:
1245:
1233:
1219:
1126:
1120:
134:
128:
119:
113:
97:
91:
13:
1:
17098:Integro-differential equation
16972:Partial differential equation
16416:(5th ed.), Brooks Cole,
16093:; Stromberg, Karl R. (1965),
16007:The Cartoon Guide to Calculus
15962:Georgiev, Svetlin G. (2018),
15891:(6th ed.), Brooks Cole,
15459:
14902:Choudary & Niculescu 2014
14678:{\textstyle {\frac {du}{dx}}}
14112:. This definition is used in
13850:by a vector space called its
11471:is a real-valued function on
10923:. If all partial derivatives
10043:with respect to the variable
10036:{\displaystyle f(x,y,\dots )}
9965:{\displaystyle \mathbf {y} '}
9227:
8916:{\displaystyle \exp(x)=e^{x}}
8756:and the third term using the
7816:{\displaystyle (fg)'=f'g+fg'}
7446:Rules for combined functions
4762:, read as "the derivative of
3124:{\displaystyle 1+1+\cdots +1}
1964:
1824:, read as "the derivative of
1102:
1097:
434:Integral of inverse functions
18:Differentiation (mathematics)
17953:Linear operators in calculus
16658:
16617:Online Derivative Calculator
16367:Schwartzman, Steven (1994),
16335:Roussos, Ioannis M. (2014),
16181:Keisler, H. Jerome (2012) ,
15844:(2nd ed.), Birkhäuser,
15678:, Cambridge University Press
14842:differential (infinitesimal)
13726:{\displaystyle \mathbb {C} }
13704:{\displaystyle \mathbb {C} }
13636:by writing a complex number
13600:{\displaystyle \mathbb {C} }
13576:{\displaystyle \mathbb {C} }
13552:{\displaystyle \mathbb {C} }
13392:{\displaystyle \mathbf {a} }
13353:. This matrix is called the
13245:{\displaystyle \mathbf {v} }
13158:{\displaystyle \mathbf {v} }
13132:{\displaystyle \mathbf {a} }
13086:{\displaystyle \mathbf {a} }
13039:{\displaystyle \mathbf {v} }
12763:{\displaystyle \mathbf {h} }
12537:{\displaystyle \mathbf {a} }
12371:{\displaystyle \mathbf {a} }
12347:{\displaystyle \mathbf {v} }
12031:{\displaystyle \mathbf {v} }
11987:{\displaystyle \mathbf {x} }
11970:exist and are continuous at
11823:{\displaystyle \mathbf {x} }
11801:{\displaystyle \mathbf {v} }
9914:{\displaystyle \mathbf {y} }
9469:{\displaystyle \mathbf {y} }
8186:{\displaystyle f(x)=h(g(x))}
7357:{\displaystyle -1<x<1}
7243:{\displaystyle -1<x<1}
5279:in the symbol of a function
4614:Notation for differentiation
4540:{\displaystyle f(x)=x^{1/3}}
4036:cannot have a derivative at
2928:{\displaystyle (a+h,f(a+h))}
1531:. This is an example of the
1297:{\displaystyle \varepsilon }
1167:, if its domain contains an
1026:instantaneous rate of change
7:
17252:Generalized Stokes' theorem
17039:Integration by substitution
16577:Encyclopedia of Mathematics
15711:Calculus: A complete course
15709:Christopher, Essex (2013),
15495:Shape Optimization Problems
15388:for the Fréchet derivative.
14530:Hewitt & Stromberg 1965
14199:
6073:{\displaystyle D_{x}f(x,y)}
5775:{\displaystyle {\ddot {y}}}
5104:can be expressed using the
4607:
2052:. This function is written
1109:function of a real variable
857:Calculus on Euclidean space
275:Logarithmic differentiation
39:Derivative (disambiguation)
10:
17979:
16781:(ε, δ)-definition of limit
16173:. See the English version
16095:Real and abstract analysis
15685:Real Analysis on Intervals
15492:Azegami, Hideyuki (2020),
14410:Henle & Kleinberg 2003
13971:from the tangent space of
13514:
12153:
11444:
9986:Functions can depend upon
9979:
9921:exists for every value of
9435:
8923:, as well as the constant
7517:is constant, then for all
6323:
5788:. It is typically used in
5744:{\displaystyle {\dot {y}}}
4611:
3375:{\displaystyle f(x)=x^{2}}
2841:, specifically the points
2318:{\displaystyle f(x)=x^{2}}
2283:be the squaring function:
1533:(ε, δ)-definition of limit
36:
29:
17914:
17814:
17733:
17674:Proof that 22/7 exceeds π
17611:
17589:
17515:
17463:Gottfried Wilhelm Leibniz
17433:
17410:e (mathematical constant)
17395:
17267:
17174:
17106:
16987:
16789:
16744:
16666:
16338:Improper Riemann Integral
16299:10.1007/978-3-319-59993-9
16271:10.1007/978-0-387-21752-9
16170:10.21136/CPMF.1922.109021
16140:10.21136/CPMF.1922.121916
16103:10.1007/978-3-662-29794-0
16056:10.1007/978-1-4612-0035-2
15974:10.1007/978-3-319-73954-0
15917:10.1007/978-3-540-46783-0
15850:10.1007/978-0-8176-8418-1
15814:10.1007/978-981-99-2935-1
15756:10.1007/978-1-4613-8955-2
15693:10.1007/978-81-322-2148-7
15674:Carothers, N. L. (2000),
15504:10.1007/978-981-15-7618-8
15233:Mathai & Haubold 2017
14888:, p. 125–126;
14256:, p. 129–130;
10970:are defined at the point
10257:partial derivative symbol
9389:infinitely differentiable
8947:Higher-order derivatives
6343:Rules for basic functions
5930:, although it seems that
5917:{\displaystyle D^{n}f(x)}
5180:{\displaystyle y=f(g(x))}
4624:Gottfried Wilhelm Leibniz
4547:is not differentiable at
3996:greater than or equal to
1041:Gottfried Wilhelm Leibniz
591:Summand limit (term test)
17425:Stirling's approximation
16898:Implicit differentiation
16846:Rules of differentiation
16221:Kreyszig, Erwin (1991),
16044:Guzman, Alberto (2003),
15905:Funaro, Daniele (1992),
15797:10.15673/tmgc.v11i2.1028
15713:, Pearson, p. 682,
15654:Cajori, Florian (2007),
15595:Bhardwaj, R. S. (2005),
15092:, p. 125–126.
15076:, p. 108–109.
14956:, p. 125–126.
14844:for further information.
14585:, p. 125–126.
14236:
14011:to the tangent space of
13916:{\displaystyle f:M\to N}
13797:Cauchy–Riemann equations
12955:is a vector starting at
11316:{\displaystyle \nabla f}
10063:is variously denoted by
9081:, and the derivative of
8952:Higher order derivatives
7576:{\displaystyle f'(x)=0.}
6839:{\displaystyle x,a>0}
5974:{\displaystyle u=f(x,y)}
4303:{\displaystyle f(x)=|x|}
2875:{\displaystyle (a,f(a))}
2710:, drawn in black, and a
2683:{\displaystyle f'(x)=2x}
1772:"; or it can be denoted
270:Implicit differentiation
260:Differentiation notation
187:Inverse function theorem
30:Not to be confused with
17659:Euler–Maclaurin formula
17564:trigonometric functions
17017:Constant of integration
16469:(Thirteenth ed.).
16391:, Courier Corporation,
16359:Rychlík, Karel (1923),
16201:Kolchin, Ellis (1973),
15996:, The MacMillan Company
15990:Goodman, A. W. (1963),
15639:2027/mdp.39015017345896
15558:Barbeau, E. J. (1961).
15114:Debnath & Shah 2015
14870:Moore & Siegel 2013
14566:Moore & Siegel 2013
11441:Directional derivatives
9447:Vector-valued functions
9355:differentiability class
9287:. A function that has
9278:{\displaystyle f^{(n)}}
8839:{\displaystyle \sin(x)}
7928:{\displaystyle \alpha }
7718:{\displaystyle \alpha }
6852:Trigonometric functions
5612:{\displaystyle f^{(n)}}
5577:{\displaystyle f^{(4)}}
2528:{\displaystyle h\neq 0}
1582:, that is if the limit
1401:{\displaystyle h\neq 0}
1319:{\displaystyle \delta }
733:Helmholtz decomposition
17948:Functions and mappings
17781:Calculus of variations
17754:Differential equations
17628:Differential geometry
17473:Infinitesimal calculus
17176:Multivariable calculus
17124:Directional derivative
16930:Second derivative test
16908:Logarithmic derivative
16881:General Leibniz's rule
16776:Order of approximation
16433:; et al. (2023),
16119:Jašek, Martin (1922),
16076:, Dover Publications,
16074:Infinitesimal Calculus
15804:Davvaz, Bijan (2023),
15579:10.4153/CMB-1961-013-0
15550:10.4064/sm-3-1-174-179
15047:
15023:
14999:
14876:, p. 78–79.
14832:
14806:
14778:
14725:
14702:
14679:
14642:
14611:
14104:
14082:
14056:
14025:
14005:
13985:
13959:
13937:
13917:
13883:
13844:
13824:
13785:
13756:
13727:
13705:
13681:
13650:
13630:
13601:
13577:
13553:
13502:
13393:
13371:
13341:
13268:
13246:
13222:
13202:
13159:
13133:
13109:
13087:
13060:
13040:
13014:
12971:
12949:
12929:
12898:
12867:
12826:
12795:
12764:
12742:
12604:
12538:
12516:
12496:
12460:
12372:
12348:
12326:
12304:
12303:{\displaystyle n>1}
12276:
12256:
12234:
12203:
12174:
12141:
12096:
12032:
12010:
11988:
11964:
11942:
11824:
11802:
11780:
11764:directional derivative
11754:
11690:
11662:
11642:
11622:
11602:
11580:
11558:
11538:
11518:
11496:
11465:
11447:Directional derivative
11427:
11369:
11317:
11299:vector-valued function
11291:
11271:
11251:
11227:
11018:
10964:
10937:
10913:
10852:
10619:
10567:
10540:
10485:
10391:
10371:
10351:
10329:
10241:
10218:
10181:
10146:
10116:
10115:{\displaystyle f'_{x}}
10086:
10057:
10037:
9988:more than one variable
9966:
9937:
9915:
9893:
9781:
9750:
9721:
9688:
9588:
9502:
9470:
9453:vector-valued function
9442:Multivariable calculus
9379:
9345:
9322:
9301:
9279:
9244:
9217:
9191:
9155:
9131:
9100:
9073:
9042:
9015:
8988:
8968:
8937:
8917:
8872:
8871:{\displaystyle \ln(x)}
8840:
8808:
8781:
8746:
8390:
8277:
8187:
8111:
8091:
8071:
7975:
7935:is a constant because
7929:
7909:
7859:
7837:
7817:
7741:
7740:{\displaystyle \beta }
7719:
7699:
7679:
7659:
7577:
7533:
7511:
7484:
7464:
7434:
7358:
7323:
7244:
7209:
7123:
6985:
6916:
6840:
6808:
6723:
6722:{\displaystyle x>0}
6697:
6636:
6635:{\displaystyle a>0}
6610:
6539:
6463:
6381:
6367:is a real number, and
6361:
6308:
6189:
6074:
6027:
6026:{\displaystyle D_{x}u}
5997:
5975:
5918:
5877:
5857:
5823:
5790:differential equations
5776:
5745:
5714:
5692:
5659:
5635:
5613:
5578:
5543:
5505:
5478:
5451:
5427:
5398:
5374:
5350:
5304:
5267:
5181:
5137:
5136:{\displaystyle u=g(x)}
5093:
4980:
4979:{\displaystyle y=f(x)}
4945:
4925:
4874:
4798:
4776:
4754:
4709:
4708:{\displaystyle y=f(x)}
4672:
4647:
4567:
4541:
4487:
4459:
4434:
4414:
4394:
4374:
4354:
4332:
4304:
4254:
4228:
4208:
4182:
4162:
4142:
4116:
4096:
4070:
4050:
4030:
4010:
3990:
3968:
3946:
3922:
3902:
3882:
3858:
3836:
3810:
3742:
3376:
3338:standard part function
3330:
3308:
3283:
3182:
3153:
3125:
3071:
3053:is as the ratio of an
3047:
2993:
2973:
2949:
2929:
2876:
2833:
2805:
2684:
2639:
2616:
2596:
2571:
2549:
2529:
2503:
2319:
2277:
2254:
2232:
2210:
2176:
2142:
2122:
2100:
2071:
2046:
2026:
2006:
1982:
1954:
1932:
1907:
1880:
1858:
1838:
1816:
1764:
1740:
1716:
1680:
1660:
1640:
1620:
1596:
1574:
1552:
1521:
1437:
1436:{\displaystyle f(a+h)}
1402:
1376:
1340:
1320:
1298:
1273:
1187:
1157:
1133:
867:Limit of distributions
687:Directional derivative
343:Faà di Bruno's formula
141:
17943:Differential calculus
17938:Mathematical analysis
17874:Representation theory
17833:quaternionic analysis
17829:Hypercomplex analysis
17727:mathematical analysis
17547:logarithmic functions
17542:exponential functions
17458:Generality of algebra
17336:Tests of convergence
16962:Differential equation
16946:Further applications
16935:Extreme value theorem
16925:First derivative test
16819:Differential operator
16791:Differential calculus
16524:Pearson Prentice Hall
16498:(September 8, 1998),
16496:Thompson, Silvanus P.
16454:Thomas, George B. Jr.
16408:(December 24, 2002),
16261:Lee, John M. (2013),
16223:Differential Geometry
15742:(December 22, 1998),
15618:Annals of Mathematics
15048:
15024:
15000:
14833:
14807:
14788:non-standard analysis
14779:
14726:
14703:
14680:
14643:
14612:
14221:Functional derivative
14185:arithmetic derivative
14114:differential geometry
14105:
14083:
14057:
14026:
14006:
13986:
13960:
13938:
13918:
13884:
13845:
13825:
13801:holomorphic functions
13786:
13757:
13728:
13706:
13682:
13651:
13631:
13602:
13578:
13554:
13503:
13394:
13372:
13342:
13269:
13247:
13223:
13203:
13160:
13134:
13110:
13088:
13061:
13041:
13015:
12972:
12950:
12930:
12899:
12868:
12827:
12796:
12765:
12743:
12605:
12539:
12517:
12497:
12461:
12373:
12349:
12327:
12305:
12277:
12257:
12235:
12204:
12175:
12142:
12076:
12033:
12011:
11989:
11965:
11943:
11825:
11803:
11781:
11755:
11691:
11663:
11643:
11623:
11603:
11581:
11559:
11539:
11519:
11497:
11466:
11428:
11370:
11318:
11292:
11272:
11252:
11228:
11019:
10965:
10963:{\displaystyle x_{j}}
10938:
10914:
10853:
10620:
10568:
10566:{\displaystyle x_{i}}
10541:
10486:
10392:
10372:
10352:
10330:
10242:
10219:
10182:
10147:
10117:
10087:
10085:{\displaystyle f_{x}}
10058:
10038:
10000:differential geometry
9967:
9938:
9916:
9894:
9787:is defined to be the
9782:
9751:
9722:
9689:
9589:
9503:
9471:
9380:
9378:{\displaystyle C^{k}}
9346:
9323:
9302:
9280:
9245:
9218:
9199:th derivative or the
9192:
9190:{\displaystyle (n-1)}
9156:
9132:
9101:
9074:
9043:
9016:
8989:
8969:
8938:
8918:
8873:
8841:
8809:
8807:{\displaystyle x^{4}}
8782:
8780:{\displaystyle x^{2}}
8747:
8391:
8278:
8188:
8112:
8092:
8072:
7981:by the constant rule.
7976:
7930:
7910:
7860:
7838:
7818:
7742:
7720:
7705:and all real numbers
7700:
7680:
7660:
7578:
7534:
7512:
7485:
7465:
7435:
7359:
7324:
7245:
7210:
7124:
6986:
6917:
6841:
6809:
6724:
6698:
6637:
6611:
6540:
6464:
6400:Derivatives of powers
6382:
6362:
6326:Differentiation rules
6309:
6190:
6075:
6028:
5998:
5976:
5919:
5878:
5858:
5856:{\displaystyle Df(x)}
5824:
5798:differential geometry
5777:
5746:
5715:
5693:
5660:
5636:
5614:
5579:
5544:
5506:
5479:
5452:
5428:
5399:
5375:
5351:
5349:{\displaystyle f'(x)}
5318:Joseph-Louis Lagrange
5305:
5268:
5182:
5138:
5094:
4981:
4946:
4926:
4875:
4808:differential operator
4799:
4777:
4755:
4710:
4673:
4648:
4568:
4542:
4488:
4460:
4435:
4415:
4395:
4375:
4355:
4333:
4305:
4255:
4229:
4209:
4183:
4163:
4143:
4117:
4097:
4071:
4051:
4031:
4011:
3991:
3969:
3947:
3923:
3903:
3883:
3859:
3837:
3811:
3743:
3382:as an example again,
3377:
3331:
3309:
3284:
3183:
3154:
3126:
3081:is a way of treating
3072:
3048:
2994:
2974:
2950:
2930:
2877:
2834:
2806:
2685:
2640:
2617:
2597:
2572:
2550:
2530:
2504:
2320:
2278:
2255:
2233:
2211:
2209:{\displaystyle f'(a)}
2177:
2175:{\displaystyle f'(a)}
2143:
2123:
2101:
2072:
2047:
2027:
2007:
1983:
1955:
1933:
1908:
1881:
1859:
1839:
1817:
1765:
1741:
1717:
1715:{\displaystyle f'(a)}
1681:
1661:
1641:
1621:
1597:
1575:
1558:is differentiable at
1553:
1522:
1438:
1403:
1377:
1341:
1326:such that, for every
1321:
1299:
1274:
1188:
1158:
1134:
1072:linear transformation
1018:graph of the function
951:Mathematical analysis
862:Generalized functions
547:arithmetico-geometric
388:Leibniz integral rule
142:
17806:Table of derivatives
17612:Miscellaneous topics
17552:hyperbolic functions
17537:irrational functions
17415:Exponential function
17268:Sequences and series
17034:Integration by parts
16456:; Weir, Maurice D.;
16151:"O funkci Bolzanově"
15037:
15013:
14989:
14822:
14793:
14737:
14712:
14689:
14652:
14629:
14598:
14368:, pp. 34, 104;
14206:Covariant derivative
14155:differential algebra
14094:
14072:
14055:{\displaystyle f(x)}
14037:
14015:
13995:
13975:
13949:
13927:
13895:
13864:
13834:
13814:
13766:
13737:
13715:
13693:
13680:{\displaystyle x+iy}
13662:
13640:
13611:
13589:
13565:
13541:
13523:linear approximation
13403:
13381:
13361:
13280:
13258:
13234:
13212:
13171:
13147:
13121:
13099:
13075:
13050:
13028:
12983:
12961:
12939:
12910:
12879:
12836:
12807:
12776:
12752:
12614:
12548:
12526:
12506:
12470:
12384:
12360:
12336:
12316:
12288:
12266:
12246:
12215:
12184:
12164:
12042:
12020:
12000:
11976:
11954:
11834:
11812:
11790:
11786:in the direction of
11770:
11704:
11674:
11652:
11632:
11612:
11592:
11570:
11548:
11528:
11508:
11477:
11455:
11379:
11327:
11323:that maps the point
11304:
11281:
11261:
11241:
11233:which is called the
11030:
10976:
10947:
10927:
10921:real-valued function
10868:
10629:
10577:
10550:
10495:
10401:
10381:
10361:
10341:
10269:
10231:
10191:
10156:
10126:
10096:
10069:
10047:
10006:
9949:
9927:
9903:
9799:
9760:
9731:
9702:
9598:
9512:
9483:
9458:
9362:
9335:
9328:times differentiable
9312:
9291:
9256:
9234:
9207:
9201:derivative of order
9169:
9145:
9130:{\displaystyle f'''}
9116:
9085:
9058:
9027:
9023:. The derivative of
9000:
8978:
8958:
8927:
8882:
8850:
8818:
8791:
8764:
8400:
8297:
8201:
8144:
8117:at all inputs where
8101:
8081:
7997:
7939:
7919:
7871:
7849:
7827:
7765:
7731:
7709:
7689:
7669:
7598:
7547:
7523:
7501:
7474:
7454:
7369:
7333:
7255:
7219:
7144:
6996:
6927:
6861:
6818:
6734:
6707:
6647:
6620:
6550:
6494:
6409:
6371:
6351:
6320:Rules of computation
6201:
6086:
6039:
6007:
5987:
5944:
5889:
5867:
5835:
5813:
5803:Another notation is
5757:
5726:
5704:
5682:
5649:
5625:
5590:
5555:
5521:
5504:{\displaystyle f'''}
5490:
5463:
5441:
5412:
5388:
5364:
5326:
5303:{\displaystyle f(x)}
5285:
5191:
5147:
5112:
4990:
4955:
4935:
4884:
4814:
4788:
4766:
4725:
4684:
4659:
4634:
4594:Weierstrass function
4580:point. Early in the
4551:
4501:
4471:
4446:
4424:
4404:
4384:
4364:
4344:
4316:
4269:
4238:
4218:
4192:
4172:
4152:
4126:
4106:
4080:
4060:
4040:
4020:
4000:
3980:
3958:
3936:
3912:
3892:
3872:
3848:
3826:
3800:
3386:
3344:
3320:
3295:
3192:
3181:{\displaystyle f(x)}
3163:
3143:
3137:nonstandard analysis
3097:
3061:
3011:
3003:Using infinitesimals
2983:
2963:
2939:
2886:
2845:
2823:
2734:
2651:
2626:
2606:
2583:
2561:
2539:
2513:
2329:
2287:
2267:
2244:
2222:
2186:
2152:
2132:
2112:
2090:
2056:
2036:
2016:
1996:
1972:
1944:
1919:
1894:
1870:
1848:
1828:
1778:
1754:
1730:
1692:
1670:
1650:
1630:
1610:
1586:
1564:
1542:
1447:
1412:
1386:
1350:
1330:
1310:
1288:
1202:
1177:
1147:
1132:{\displaystyle f(x)}
1114:
1088:real-valued function
1022:linear approximation
956:Nonstandard analysis
424:Lebesgue integration
294:Rules and identities
65:
37:For other uses, see
17886:Continuous function
17839:Functional analysis
17599:List of derivatives
17435:History of calculus
17350:Cauchy condensation
17247:Exterior derivative
17204:Lagrange multiplier
16940:Maximum and minimum
16771:Limit of a sequence
16759:Limit of a function
16706:Graph of a function
16686:Continuous function
15946:2011mmop.book.....G
15352:, pp. 261–264.
14814:exterior derivative
14396:, pp. 902–904.
14216:Exterior derivative
14193:prime factorization
14178:time scale calculus
13607:is identified with
11689:{\displaystyle y=x}
10397:are, respectively:
10111:
9976:Partial derivatives
9432:In other dimensions
9099:{\displaystyle f''}
9072:{\displaystyle f''}
8289:Computation example
8136:composite functions
6219:
5477:{\displaystyle f''}
5312:. This is known as
4582:history of calculus
4566:{\displaystyle x=0}
4495:tangent is vertical
4486:{\displaystyle x=0}
4331:{\displaystyle x=0}
4253:{\displaystyle a+h}
4207:{\displaystyle a+h}
4141:{\displaystyle a+h}
4095:{\displaystyle a+h}
2708:graph of a function
2079:derivative function
1084:partial derivatives
627:Cauchy condensation
429:Contour integration
155:Fundamental theorem
82:
17918:Mathematics portal
17801:Lists of integrals
17532:rational functions
17499:Method of Fluxions
17345:Alternating series
17242:Differential forms
17224:Partial derivative
17184:Divergence theorem
17066:Quadratic integral
16834:Leibniz's notation
16824:Mean value theorem
16809:Partial derivative
16754:Indeterminate form
16600:Weisstein, Eric W.
16501:Calculus Made Easy
16436:Calculus, volume 1
16207:, Academic Press,
16121:"Funkce Bolzanova"
16010:, William Morrow,
15687:, Springer India,
15043:
15019:
14995:
14828:
14805:{\displaystyle du}
14802:
14774:
14773:
14724:{\displaystyle du}
14721:
14701:{\displaystyle dx}
14698:
14675:
14641:{\displaystyle du}
14638:
14610:{\displaystyle du}
14607:
14478:Strang et al. 2023
14474:Thomas et al. 2014
14454:Strang et al. 2023
14450:Thomas et al. 2014
14430:Strang et al. 2023
14426:Thomas et al. 2014
14338:Strang et al. 2023
14318:Strang et al. 2023
14302:Thomas et al. 2014
14282:Strang et al. 2023
14258:Strang et al. 2023
14174:finite differences
14133:Fréchet derivative
14129:Gateaux derivative
14100:
14078:
14052:
14021:
14001:
13981:
13955:
13933:
13913:
13879:
13840:
13820:
13781:
13752:
13723:
13701:
13677:
13646:
13626:
13597:
13573:
13549:
13498:
13389:
13367:
13337:
13264:
13242:
13218:
13198:
13155:
13129:
13105:
13083:
13056:
13036:
13010:
12967:
12945:
12925:
12894:
12863:
12822:
12791:
12760:
12738:
12634:
12600:
12534:
12512:
12492:
12456:
12368:
12344:
12322:
12300:
12272:
12252:
12230:
12199:
12170:
12137:
12028:
12006:
11984:
11960:
11938:
11883:
11820:
11798:
11776:
11750:
11686:
11658:
11638:
11618:
11598:
11576:
11554:
11534:
11514:
11492:
11461:
11423:
11365:
11313:
11287:
11267:
11247:
11223:
11014:
10960:
10933:
10909:
10848:
10712:
10625:is defined to be:
10615:
10563:
10536:
10481:
10387:
10367:
10347:
10325:
10237:
10214:
10177:
10142:
10112:
10099:
10082:
10053:
10033:
9992:partial derivative
9982:Partial derivative
9962:
9933:
9911:
9889:
9839:
9777:
9746:
9717:
9684:
9584:
9498:
9466:
9375:
9341:
9318:
9297:
9275:
9250:may be denoted as
9240:
9213:
9187:
9151:
9127:
9096:
9069:
9041:{\displaystyle f'}
9038:
9014:{\displaystyle f'}
9011:
8984:
8964:
8943:, were also used.
8933:
8913:
8868:
8836:
8804:
8777:
8742:
8740:
8386:
8273:
8183:
8107:
8087:
8077:for all functions
8067:
7971:
7925:
7905:
7855:
7833:
7823:for all functions
7813:
7737:
7715:
7695:
7675:
7665:for all functions
7655:
7573:
7529:
7507:
7480:
7460:
7430:
7354:
7319:
7240:
7205:
7119:
6981:
6912:
6836:
6804:
6719:
6693:
6632:
6606:
6535:
6459:
6377:
6357:
6304:
6303:
6205:
6185:
6070:
6023:
5993:
5971:
5914:
5883:-th derivative is
5873:
5853:
5819:
5772:
5741:
5710:
5688:
5655:
5631:
5609:
5574:
5539:
5501:
5474:
5447:
5426:{\displaystyle y'}
5423:
5394:
5370:
5346:
5300:
5263:
5177:
5133:
5089:
4976:
4951:-th derivative of
4941:
4921:
4870:
4794:
4772:
4750:
4749:
4705:
4671:{\displaystyle dx}
4668:
4646:{\displaystyle dy}
4643:
4590:Lipschitz function
4563:
4537:
4483:
4458:{\displaystyle -1}
4455:
4430:
4410:
4390:
4370:
4350:
4328:
4300:
4265:function given by
4250:
4224:
4204:
4188:is positive, then
4178:
4158:
4148:is very steep; as
4138:
4112:
4092:
4076:is negative, then
4066:
4046:
4026:
4006:
3986:
3964:
3942:
3918:
3898:
3878:
3854:
3832:
3806:
3768:jump discontinuity
3738:
3736:
3372:
3326:
3307:{\displaystyle dx}
3304:
3279:
3178:
3149:
3121:
3067:
3043:
2989:
2969:
2945:
2925:
2872:
2829:
2801:
2680:
2638:{\displaystyle 2a}
2635:
2612:
2595:{\displaystyle 2a}
2592:
2567:
2545:
2525:
2499:
2315:
2273:
2250:
2228:
2206:
2172:
2138:
2118:
2096:
2077:and is called the
2070:{\displaystyle f'}
2067:
2042:
2022:
2002:
1978:
1950:
1931:{\displaystyle dx}
1928:
1906:{\displaystyle df}
1903:
1876:
1854:
1834:
1812:
1811:
1760:
1736:
1712:
1676:
1656:
1636:
1616:
1592:
1570:
1548:
1517:
1433:
1398:
1372:
1336:
1316:
1294:
1269:
1226:
1183:
1153:
1129:
799:Partial derivative
728:generalized Stokes
622:Alternating series
503:Reduction formulae
492:Heaviside's method
473:tangent half-angle
460:Cylindrical shells
383:Integral transform
378:Lists of integrals
182:Mean value theorem
137:
68:
17925:
17924:
17891:Special functions
17854:Harmonic analysis
17692:
17691:
17618:Complex calculus
17607:
17606:
17488:Law of Continuity
17420:Natural logarithm
17405:Bernoulli numbers
17396:Special functions
17355:Direct comparison
17219:Multiple integral
17093:Integral equation
16989:Integral calculus
16920:Stationary points
16894:Other techniques
16839:Newton's notation
16804:Second derivative
16696:Finite difference
16556:978-0-387-90894-6
16511:978-0-312-18548-0
16463:Thomas's Calculus
16446:978-1-947172-13-5
16423:978-0-534-39339-7
16352:978-1-4665-8807-3
16328:978-0-691-15995-9
16308:978-3-319-59993-9
16280:978-0-387-21752-9
16254:978-0-618-60624-5
16214:978-0-08-087369-5
16194:978-0-871-50911-6
16112:978-3-662-28275-5
16083:978-0-486-42886-4
16065:978-1-4612-0035-2
16037:978-1-58488-448-4
16017:978-0-06-168909-3
15983:978-3-319-73954-0
15955:978-1-139-49269-0
15926:978-3-540-46783-0
15898:978-0-03-029558-4
15859:978-0-8176-8418-1
15823:978-981-99-2935-1
15765:978-3-540-65058-4
15702:978-81-322-2148-7
15667:978-1-60206-713-4
15513:978-981-15-7618-8
15485:978-0-471-00005-1
15251:, pp. 36–37.
15046:{\displaystyle a}
15022:{\displaystyle a}
14998:{\displaystyle e}
14831:{\displaystyle u}
14673:
14384:, pp. 84–85.
14356:, pp. 77–80.
14103:{\displaystyle N}
14081:{\displaystyle M}
14024:{\displaystyle N}
14004:{\displaystyle x}
13984:{\displaystyle M}
13958:{\displaystyle M}
13936:{\displaystyle x}
13843:{\displaystyle x}
13823:{\displaystyle M}
13649:{\displaystyle z}
13535:complex variables
13531:complex functions
13480:
13370:{\displaystyle f}
13267:{\displaystyle f}
13228:in the direction
13221:{\displaystyle f}
13108:{\displaystyle f}
13059:{\displaystyle f}
12970:{\displaystyle a}
12948:{\displaystyle v}
12730:
12617:
12515:{\displaystyle f}
12325:{\displaystyle f}
12275:{\displaystyle f}
12255:{\displaystyle f}
12173:{\displaystyle f}
12132:
12016:in the direction
12009:{\displaystyle f}
11963:{\displaystyle f}
11933:
11868:
11779:{\displaystyle f}
11661:{\displaystyle f}
11641:{\displaystyle y}
11621:{\displaystyle x}
11601:{\displaystyle f}
11579:{\displaystyle y}
11557:{\displaystyle x}
11544:is a function of
11537:{\displaystyle f}
11517:{\displaystyle f}
11464:{\displaystyle f}
11290:{\displaystyle f}
11270:{\displaystyle a}
11250:{\displaystyle f}
11178:
11107:
10936:{\displaystyle f}
10843:
10697:
10657:
10546:in the direction
10461:
10422:
10390:{\displaystyle y}
10370:{\displaystyle x}
10350:{\displaystyle f}
10247:-direction. Here
10240:{\displaystyle x}
10212:
10172:
10056:{\displaystyle x}
9936:{\displaystyle t}
9884:
9824:
9696:parametric curves
9403:constant function
9344:{\displaystyle k}
9321:{\displaystyle k}
9300:{\displaystyle k}
9243:{\displaystyle f}
9216:{\displaystyle n}
9154:{\displaystyle n}
9050:second derivative
8987:{\displaystyle f}
8967:{\displaystyle f}
8936:{\displaystyle 7}
8695:
8620:
8557:
8492:
8110:{\displaystyle g}
8090:{\displaystyle f}
8065:
8013:
7858:{\displaystyle g}
7836:{\displaystyle f}
7698:{\displaystyle g}
7678:{\displaystyle f}
7532:{\displaystyle x}
7510:{\displaystyle f}
7483:{\displaystyle g}
7463:{\displaystyle f}
7428:
7385:
7317:
7316:
7271:
7203:
7202:
7160:
7086:
7012:
6943:
6877:
6802:
6750:
6691:
6663:
6566:
6510:
6486:with general base
6480:natural logarithm
6425:
6380:{\displaystyle e}
6360:{\displaystyle a}
6276:
6254:
6158:
6136:
5996:{\displaystyle x}
5876:{\displaystyle n}
5822:{\displaystyle D}
5769:
5738:
5713:{\displaystyle t}
5698:is a function of
5691:{\displaystyle y}
5672:Newton's notation
5658:{\displaystyle f}
5643:th derivative of
5634:{\displaystyle n}
5450:{\displaystyle y}
5397:{\displaystyle x}
5373:{\displaystyle f}
5258:
5235:
5212:
5102:composed function
5065:
5043:
5025:
4944:{\displaystyle n}
4919:
4853:
4835:
4797:{\displaystyle x}
4775:{\displaystyle y}
4747:
4433:{\displaystyle h}
4413:{\displaystyle 0}
4393:{\displaystyle h}
4373:{\displaystyle h}
4353:{\displaystyle h}
4310:is continuous at
4227:{\displaystyle a}
4181:{\displaystyle h}
4161:{\displaystyle h}
4115:{\displaystyle a}
4069:{\displaystyle h}
4049:{\displaystyle a}
4029:{\displaystyle f}
4009:{\displaystyle a}
3989:{\displaystyle x}
3967:{\displaystyle a}
3945:{\displaystyle x}
3921:{\displaystyle f}
3901:{\displaystyle a}
3881:{\displaystyle a}
3857:{\displaystyle f}
3835:{\displaystyle a}
3809:{\displaystyle f}
3667:
3631:
3577:
3502:
3273:
3152:{\displaystyle d}
3079:hyperreal numbers
3070:{\displaystyle f}
3032:
2992:{\displaystyle a}
2972:{\displaystyle f}
2948:{\displaystyle h}
2832:{\displaystyle f}
2615:{\displaystyle a}
2570:{\displaystyle 0}
2548:{\displaystyle h}
2479:
2419:
2372:
2276:{\displaystyle f}
2263:For example, let
2253:{\displaystyle f}
2231:{\displaystyle f}
2141:{\displaystyle a}
2121:{\displaystyle f}
2099:{\displaystyle f}
2045:{\displaystyle x}
2025:{\displaystyle f}
2005:{\displaystyle x}
1981:{\displaystyle f}
1953:{\displaystyle a}
1879:{\displaystyle a}
1857:{\displaystyle x}
1837:{\displaystyle f}
1800:
1763:{\displaystyle a}
1739:{\displaystyle f}
1679:{\displaystyle a}
1659:{\displaystyle f}
1639:{\displaystyle a}
1619:{\displaystyle f}
1595:{\displaystyle L}
1573:{\displaystyle a}
1551:{\displaystyle f}
1501:
1443:is defined, and
1339:{\displaystyle h}
1267:
1211:
1186:{\displaystyle a}
1156:{\displaystyle a}
994:
993:
874:
873:
836:
835:
804:Multiple integral
740:
739:
644:
643:
611:Direct comparison
582:Convergence tests
520:
519:
488:Partial fractions
355:
354:
265:Second derivative
16:(Redirected from
17970:
17844:Fourier analysis
17824:Complex analysis
17725:Major topics in
17719:
17712:
17705:
17696:
17695:
17622:Contour integral
17520:
17519:
17370:Limit comparison
17279:Types of series
17238:Advanced topics
17229:Surface integral
17073:Trapezoidal rule
17012:Basic properties
17007:Riemann integral
16955:Taylor's theorem
16681:Concave function
16676:Binomial theorem
16653:
16646:
16639:
16630:
16629:
16613:
16612:
16585:
16559:
16541:Warner, Frank W.
16536:
16522:(9th ed.),
16514:
16491:
16489:
16487:
16468:
16449:
16426:
16415:
16401:
16383:
16363:
16355:
16331:
16311:
16283:
16257:
16239:
16217:
16197:
16172:
16155:
16142:
16125:
16115:
16086:
16068:
16040:
16020:
15997:
15986:
15958:
15929:
15901:
15883:
15862:
15834:
15800:
15799:
15789:
15768:
15736:Courant, Richard
15731:
15705:
15679:
15670:
15650:
15641:
15609:
15591:
15581:
15554:
15552:
15524:
15488:
15477:
15453:
15447:
15441:
15431:
15425:
15411:
15405:
15395:
15389:
15375:
15369:
15359:
15353:
15347:
15341:
15335:
15329:
15319:
15313:
15307:
15301:
15291:
15280:
15270:
15264:
15258:
15252:
15246:
15240:
15230:
15224:
15206:
15200:
15190:
15184:
15181:Christopher 2013
15170:
15164:
15158:
15149:
15143:
15137:
15127:
15121:
15111:
15105:
15099:
15093:
15083:
15077:
15062:
15056:
15054:
15052:
15050:
15049:
15044:
15030:
15028:
15026:
15025:
15020:
15006:
15004:
15002:
15001:
14996:
14978:
14972:
14966:
14957:
14947:
14941:
14935:
14929:
14919:
14913:
14899:
14893:
14883:
14877:
14867:
14861:
14854:Schwartzman 1994
14851:
14845:
14839:
14837:
14835:
14834:
14829:
14811:
14809:
14808:
14803:
14785:
14783:
14781:
14780:
14775:
14763:
14730:
14728:
14727:
14722:
14707:
14705:
14704:
14699:
14685:. Others define
14684:
14682:
14681:
14676:
14674:
14672:
14664:
14656:
14647:
14645:
14644:
14639:
14616:
14614:
14613:
14608:
14592:
14586:
14580:
14569:
14563:
14557:
14551:
14545:
14539:
14533:
14523:
14517:
14511:
14505:
14491:
14485:
14467:
14461:
14443:
14437:
14419:
14413:
14403:
14397:
14391:
14385:
14379:
14373:
14363:
14357:
14351:
14345:
14331:
14325:
14311:
14305:
14295:
14289:
14271:
14265:
14247:
14111:
14109:
14107:
14106:
14101:
14087:
14085:
14084:
14079:
14063:
14061:
14059:
14058:
14053:
14030:
14028:
14027:
14022:
14010:
14008:
14007:
14002:
13990:
13988:
13987:
13982:
13966:
13964:
13962:
13961:
13956:
13942:
13940:
13939:
13934:
13922:
13920:
13919:
13914:
13890:
13888:
13886:
13885:
13880:
13878:
13877:
13872:
13849:
13847:
13846:
13841:
13829:
13827:
13826:
13821:
13790:
13788:
13787:
13782:
13780:
13779:
13774:
13761:
13759:
13758:
13753:
13751:
13750:
13745:
13732:
13730:
13729:
13724:
13722:
13710:
13708:
13707:
13702:
13700:
13688:
13686:
13684:
13683:
13678:
13655:
13653:
13652:
13647:
13635:
13633:
13632:
13627:
13625:
13624:
13619:
13606:
13604:
13603:
13598:
13596:
13584:
13582:
13580:
13579:
13574:
13572:
13558:
13556:
13555:
13550:
13548:
13507:
13505:
13504:
13499:
13494:
13493:
13485:
13481:
13479:
13478:
13477:
13464:
13463:
13462:
13449:
13439:
13438:
13437:
13421:
13413:
13398:
13396:
13395:
13390:
13388:
13376:
13374:
13373:
13368:
13348:
13346:
13344:
13343:
13338:
13333:
13332:
13314:
13313:
13301:
13300:
13273:
13271:
13270:
13265:
13253:
13251:
13249:
13248:
13243:
13241:
13227:
13225:
13224:
13219:
13207:
13205:
13204:
13199:
13197:
13189:
13181:
13166:
13164:
13162:
13161:
13156:
13154:
13140:
13138:
13136:
13135:
13130:
13128:
13114:
13112:
13111:
13106:
13094:
13092:
13090:
13089:
13084:
13082:
13065:
13063:
13062:
13057:
13045:
13043:
13042:
13037:
13035:
13019:
13017:
13016:
13011:
13009:
13001:
12993:
12978:
12976:
12974:
12973:
12968:
12954:
12952:
12951:
12946:
12934:
12932:
12931:
12926:
12924:
12923:
12918:
12905:
12903:
12901:
12900:
12895:
12893:
12892:
12887:
12872:
12870:
12869:
12864:
12862:
12854:
12846:
12831:
12829:
12828:
12823:
12821:
12820:
12815:
12802:
12800:
12798:
12797:
12792:
12790:
12789:
12784:
12769:
12767:
12766:
12761:
12759:
12747:
12745:
12744:
12739:
12731:
12729:
12725:
12716:
12709:
12701:
12693:
12679:
12659:
12651:
12636:
12633:
12626:
12609:
12607:
12606:
12601:
12599:
12598:
12593:
12584:
12583:
12578:
12566:
12558:
12543:
12541:
12540:
12535:
12533:
12521:
12519:
12518:
12513:
12501:
12499:
12498:
12493:
12488:
12480:
12465:
12463:
12462:
12457:
12452:
12444:
12436:
12422:
12405:
12397:
12379:
12377:
12375:
12374:
12369:
12367:
12353:
12351:
12350:
12345:
12343:
12331:
12329:
12328:
12323:
12311:
12309:
12307:
12306:
12301:
12281:
12279:
12278:
12273:
12261:
12259:
12258:
12253:
12241:
12239:
12237:
12236:
12231:
12229:
12228:
12223:
12208:
12206:
12205:
12200:
12198:
12197:
12192:
12179:
12177:
12176:
12171:
12156:Total derivative
12146:
12144:
12143:
12138:
12133:
12131:
12130:
12129:
12116:
12108:
12106:
12105:
12095:
12090:
12069:
12061:
12056:
12055:
12054:
12038:by the formula:
12037:
12035:
12034:
12029:
12027:
12015:
12013:
12012:
12007:
11995:
11993:
11991:
11990:
11985:
11983:
11969:
11967:
11966:
11961:
11947:
11945:
11944:
11939:
11934:
11929:
11925:
11908:
11897:
11885:
11882:
11861:
11853:
11848:
11847:
11846:
11829:
11827:
11826:
11821:
11819:
11807:
11805:
11804:
11799:
11797:
11785:
11783:
11782:
11777:
11761:
11759:
11757:
11756:
11751:
11746:
11745:
11727:
11726:
11711:
11697:
11695:
11693:
11692:
11687:
11667:
11665:
11664:
11659:
11647:
11645:
11644:
11639:
11627:
11625:
11624:
11619:
11607:
11605:
11604:
11599:
11587:
11585:
11583:
11582:
11577:
11563:
11561:
11560:
11555:
11543:
11541:
11540:
11535:
11523:
11521:
11520:
11515:
11503:
11501:
11499:
11498:
11493:
11491:
11490:
11485:
11470:
11468:
11467:
11462:
11432:
11430:
11429:
11424:
11419:
11418:
11400:
11399:
11374:
11372:
11371:
11366:
11361:
11360:
11342:
11341:
11322:
11320:
11319:
11314:
11296:
11294:
11293:
11288:
11276:
11274:
11273:
11268:
11256:
11254:
11253:
11248:
11232:
11230:
11229:
11224:
11219:
11215:
11211:
11210:
11192:
11191:
11179:
11177:
11176:
11175:
11162:
11154:
11140:
11139:
11121:
11120:
11108:
11106:
11105:
11104:
11091:
11083:
11070:
11069:
11051:
11050:
11025:
11023:
11021:
11020:
11015:
11010:
11009:
10991:
10990:
10969:
10967:
10966:
10961:
10959:
10958:
10943:with respect to
10942:
10940:
10939:
10934:
10918:
10916:
10915:
10910:
10905:
10904:
10886:
10885:
10857:
10855:
10854:
10849:
10844:
10839:
10835:
10834:
10816:
10815:
10797:
10796:
10775:
10774:
10750:
10749:
10731:
10730:
10714:
10711:
10690:
10689:
10671:
10670:
10658:
10656:
10655:
10654:
10641:
10633:
10624:
10622:
10621:
10616:
10611:
10610:
10592:
10591:
10572:
10570:
10569:
10564:
10562:
10561:
10545:
10543:
10542:
10537:
10532:
10531:
10513:
10512:
10490:
10488:
10487:
10482:
10462:
10460:
10452:
10444:
10423:
10421:
10413:
10405:
10396:
10394:
10393:
10388:
10376:
10374:
10373:
10368:
10356:
10354:
10353:
10348:
10336:
10334:
10332:
10331:
10326:
10324:
10323:
10302:
10301:
10246:
10244:
10243:
10238:
10223:
10221:
10220:
10215:
10213:
10211:
10203:
10195:
10186:
10184:
10183:
10178:
10173:
10171:
10160:
10151:
10149:
10148:
10143:
10138:
10137:
10121:
10119:
10118:
10113:
10107:
10091:
10089:
10088:
10083:
10081:
10080:
10062:
10060:
10059:
10054:
10042:
10040:
10039:
10034:
9971:
9969:
9968:
9963:
9961:
9957:
9944:
9942:
9940:
9939:
9934:
9920:
9918:
9917:
9912:
9910:
9898:
9896:
9895:
9890:
9885:
9880:
9870:
9847:
9841:
9838:
9811:
9807:
9786:
9784:
9783:
9778:
9767:
9755:
9753:
9752:
9747:
9745:
9744:
9739:
9726:
9724:
9723:
9718:
9716:
9715:
9710:
9693:
9691:
9690:
9685:
9671:
9670:
9643:
9642:
9621:
9620:
9605:
9593:
9591:
9590:
9585:
9574:
9573:
9546:
9545:
9524:
9523:
9507:
9505:
9504:
9499:
9497:
9496:
9491:
9475:
9473:
9472:
9467:
9465:
9386:
9384:
9382:
9381:
9376:
9374:
9373:
9352:
9350:
9348:
9347:
9342:
9327:
9325:
9324:
9319:
9306:
9304:
9303:
9298:
9286:
9284:
9282:
9281:
9276:
9274:
9273:
9249:
9247:
9246:
9241:
9224:
9222:
9220:
9219:
9214:
9198:
9196:
9194:
9193:
9188:
9162:
9160:
9158:
9157:
9152:
9138:
9136:
9134:
9133:
9128:
9126:
9108:third derivative
9105:
9103:
9102:
9097:
9095:
9080:
9078:
9076:
9075:
9070:
9068:
9047:
9045:
9044:
9039:
9037:
9022:
9020:
9018:
9017:
9012:
9010:
8993:
8991:
8990:
8985:
8973:
8971:
8970:
8965:
8942:
8940:
8939:
8934:
8922:
8920:
8919:
8914:
8912:
8911:
8877:
8875:
8874:
8869:
8845:
8843:
8842:
8837:
8813:
8811:
8810:
8805:
8803:
8802:
8786:
8784:
8783:
8778:
8776:
8775:
8751:
8749:
8748:
8743:
8741:
8734:
8733:
8706:
8705:
8696:
8688:
8683:
8679:
8678:
8650:
8649:
8631:
8621:
8619:
8611:
8610:
8606:
8605:
8588:
8568:
8567:
8558:
8556:
8548:
8547:
8543:
8542:
8522:
8517:
8513:
8512:
8493:
8491:
8483:
8482:
8478:
8477:
8460:
8455:
8454:
8414:
8395:
8393:
8392:
8387:
8379:
8378:
8351:
8347:
8346:
8324:
8323:
8282:
8280:
8279:
8274:
8260:
8231:
8211:
8194:
8192:
8190:
8189:
8184:
8123:
8116:
8114:
8113:
8108:
8096:
8094:
8093:
8088:
8076:
8074:
8073:
8068:
8066:
8064:
8063:
8054:
8053:
8036:
8027:
8022:
8018:
8014:
8006:
7980:
7978:
7977:
7972:
7949:
7934:
7932:
7931:
7926:
7914:
7912:
7911:
7906:
7904:
7890:
7866:
7864:
7862:
7861:
7856:
7842:
7840:
7839:
7834:
7822:
7820:
7819:
7814:
7812:
7795:
7784:
7748:
7746:
7744:
7743:
7738:
7724:
7722:
7721:
7716:
7704:
7702:
7701:
7696:
7684:
7682:
7681:
7676:
7664:
7662:
7661:
7656:
7654:
7640:
7626:
7582:
7580:
7579:
7574:
7557:
7540:
7538:
7536:
7535:
7530:
7516:
7514:
7513:
7508:
7489:
7487:
7486:
7481:
7469:
7467:
7466:
7461:
7439:
7437:
7436:
7431:
7429:
7427:
7426:
7425:
7406:
7386:
7384:
7373:
7363:
7361:
7360:
7355:
7328:
7326:
7325:
7320:
7318:
7315:
7314:
7299:
7295:
7272:
7270:
7259:
7249:
7247:
7246:
7241:
7214:
7212:
7211:
7206:
7204:
7201:
7200:
7185:
7181:
7161:
7159:
7148:
7128:
7126:
7125:
7120:
7106:
7105:
7087:
7085:
7072:
7071:
7058:
7041:
7040:
7013:
7011:
7000:
6990:
6988:
6987:
6982:
6944:
6942:
6931:
6921:
6919:
6918:
6913:
6878:
6876:
6865:
6845:
6843:
6842:
6837:
6813:
6811:
6810:
6805:
6803:
6801:
6778:
6761:
6760:
6751:
6749:
6738:
6728:
6726:
6725:
6720:
6702:
6700:
6699:
6694:
6692:
6684:
6664:
6662:
6651:
6641:
6639:
6638:
6633:
6615:
6613:
6612:
6607:
6590:
6589:
6577:
6576:
6567:
6565:
6554:
6544:
6542:
6541:
6536:
6534:
6533:
6521:
6520:
6511:
6509:
6498:
6468:
6466:
6465:
6460:
6458:
6457:
6436:
6435:
6426:
6424:
6413:
6392:
6386:
6384:
6383:
6378:
6366:
6364:
6363:
6358:
6315:
6313:
6311:
6310:
6305:
6302:
6301:
6277:
6275:
6264:
6262:
6261:
6255:
6253:
6242:
6218:
6213:
6194:
6192:
6191:
6186:
6184:
6183:
6159:
6157:
6146:
6144:
6143:
6137:
6135:
6124:
6101:
6100:
6081:
6079:
6077:
6076:
6071:
6051:
6050:
6032:
6030:
6029:
6024:
6019:
6018:
6002:
6000:
5999:
5994:
5982:
5980:
5978:
5977:
5972:
5925:
5923:
5921:
5920:
5915:
5901:
5900:
5882:
5880:
5879:
5874:
5862:
5860:
5859:
5854:
5830:
5828:
5826:
5825:
5820:
5783:
5781:
5779:
5778:
5773:
5771:
5770:
5762:
5750:
5748:
5747:
5742:
5740:
5739:
5731:
5721:
5719:
5717:
5716:
5711:
5697:
5695:
5694:
5689:
5666:
5664:
5662:
5661:
5656:
5642:
5640:
5638:
5637:
5632:
5618:
5616:
5615:
5610:
5608:
5607:
5585:
5583:
5581:
5580:
5575:
5573:
5572:
5548:
5546:
5545:
5540:
5538:
5537:
5536:
5512:
5510:
5508:
5507:
5502:
5500:
5483:
5481:
5480:
5475:
5473:
5458:
5456:
5454:
5453:
5448:
5434:
5432:
5430:
5429:
5424:
5422:
5405:
5403:
5401:
5400:
5395:
5381:
5379:
5377:
5376:
5371:
5357:
5355:
5353:
5352:
5347:
5336:
5311:
5309:
5307:
5306:
5301:
5272:
5270:
5269:
5264:
5259:
5257:
5249:
5241:
5236:
5234:
5226:
5218:
5213:
5211:
5203:
5195:
5186:
5184:
5183:
5178:
5142:
5140:
5139:
5134:
5098:
5096:
5095:
5090:
5085:
5084:
5066:
5064:
5053:
5051:
5050:
5044:
5042:
5031:
5026:
5024:
5023:
5022:
5009:
5005:
5004:
4994:
4985:
4983:
4982:
4977:
4950:
4948:
4947:
4942:
4930:
4928:
4927:
4922:
4920:
4918:
4917:
4916:
4903:
4899:
4898:
4888:
4879:
4877:
4876:
4871:
4854:
4852:
4841:
4836:
4834:
4826:
4818:
4805:
4803:
4801:
4800:
4795:
4782:with respect to
4781:
4779:
4778:
4773:
4761:
4759:
4757:
4756:
4751:
4748:
4746:
4738:
4730:
4714:
4712:
4711:
4706:
4679:
4677:
4675:
4674:
4669:
4652:
4650:
4649:
4644:
4622:, introduced by
4620:Leibniz notation
4572:
4570:
4569:
4564:
4546:
4544:
4543:
4538:
4536:
4535:
4531:
4492:
4490:
4489:
4484:
4466:
4464:
4462:
4461:
4456:
4439:
4437:
4436:
4431:
4419:
4417:
4416:
4411:
4399:
4397:
4396:
4391:
4379:
4377:
4376:
4371:
4359:
4357:
4356:
4351:
4339:
4337:
4335:
4334:
4329:
4309:
4307:
4306:
4301:
4299:
4291:
4259:
4257:
4256:
4251:
4233:
4231:
4230:
4225:
4213:
4211:
4210:
4205:
4187:
4185:
4184:
4179:
4167:
4165:
4164:
4159:
4147:
4145:
4144:
4139:
4121:
4119:
4118:
4113:
4101:
4099:
4098:
4093:
4075:
4073:
4072:
4067:
4055:
4053:
4052:
4047:
4035:
4033:
4032:
4027:
4016:. The function
4015:
4013:
4012:
4007:
3995:
3993:
3992:
3987:
3975:
3973:
3971:
3970:
3965:
3951:
3949:
3948:
3943:
3927:
3925:
3924:
3919:
3907:
3905:
3904:
3899:
3887:
3885:
3884:
3879:
3863:
3861:
3860:
3855:
3843:
3841:
3839:
3838:
3833:
3815:
3813:
3812:
3807:
3788:
3779:
3763:
3747:
3745:
3744:
3739:
3737:
3718:
3714:
3710:
3677:
3673:
3669:
3668:
3666:
3658:
3657:
3656:
3637:
3632:
3630:
3622:
3605:
3586:
3582:
3578:
3576:
3568:
3567:
3566:
3529:
3511:
3507:
3503:
3501:
3493:
3492:
3491:
3479:
3478:
3439:
3438:
3428:
3400:
3381:
3379:
3378:
3373:
3371:
3370:
3335:
3333:
3332:
3327:
3315:
3313:
3311:
3310:
3305:
3288:
3286:
3285:
3280:
3278:
3274:
3272:
3264:
3226:
3202:
3187:
3185:
3184:
3179:
3158:
3156:
3155:
3150:
3130:
3128:
3127:
3122:
3076:
3074:
3073:
3068:
3052:
3050:
3049:
3044:
3033:
3031:
3023:
3015:
2998:
2996:
2995:
2990:
2978:
2976:
2975:
2970:
2959:to the graph of
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2881:
2879:
2878:
2873:
2840:
2838:
2836:
2835:
2830:
2810:
2808:
2807:
2802:
2800:
2796:
2795:
2776:
2775:
2760:
2756:
2755:
2727:
2703:
2691:
2689:
2687:
2686:
2681:
2661:
2644:
2642:
2641:
2636:
2621:
2619:
2618:
2613:
2601:
2599:
2598:
2593:
2578:
2576:
2574:
2573:
2568:
2554:
2552:
2551:
2546:
2534:
2532:
2531:
2526:
2508:
2506:
2505:
2500:
2480:
2475:
2474:
2473:
2461:
2460:
2436:
2435:
2425:
2420:
2415:
2414:
2413:
2401:
2400:
2378:
2373:
2368:
2333:
2324:
2322:
2321:
2316:
2314:
2313:
2282:
2280:
2279:
2274:
2259:
2257:
2256:
2251:
2239:
2237:
2235:
2234:
2229:
2215:
2213:
2212:
2207:
2196:
2181:
2179:
2178:
2173:
2162:
2147:
2145:
2144:
2139:
2127:
2125:
2124:
2119:
2107:
2105:
2103:
2102:
2097:
2076:
2074:
2073:
2068:
2066:
2051:
2049:
2048:
2043:
2031:
2029:
2028:
2023:
2011:
2009:
2008:
2003:
1987:
1985:
1984:
1979:
1961:
1959:
1957:
1956:
1951:
1937:
1935:
1934:
1929:
1914:
1912:
1910:
1909:
1904:
1887:
1885:
1883:
1882:
1877:
1863:
1861:
1860:
1855:
1844:with respect to
1843:
1841:
1840:
1835:
1823:
1821:
1819:
1818:
1813:
1801:
1799:
1791:
1783:
1771:
1769:
1767:
1766:
1761:
1747:
1745:
1743:
1742:
1737:
1723:
1721:
1719:
1718:
1713:
1702:
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1645:
1643:
1642:
1637:
1625:
1623:
1622:
1617:
1601:
1599:
1598:
1593:
1581:
1579:
1577:
1576:
1571:
1557:
1555:
1554:
1549:
1538:If the function
1526:
1524:
1523:
1518:
1507:
1503:
1502:
1497:
1462:
1442:
1440:
1439:
1434:
1407:
1405:
1404:
1399:
1381:
1379:
1378:
1373:
1365:
1357:
1345:
1343:
1342:
1337:
1325:
1323:
1322:
1317:
1305:
1303:
1301:
1300:
1295:
1278:
1276:
1275:
1270:
1268:
1263:
1228:
1225:
1194:
1192:
1190:
1189:
1184:
1162:
1160:
1159:
1154:
1138:
1136:
1135:
1130:
1055:is the object's
1037:Leibniz notation
986:
979:
972:
920:
885:
851:
850:
847:
814:Surface integral
757:
756:
753:
661:
660:
657:
617:Limit comparison
537:
536:
533:
419:Riemann integral
372:
371:
368:
328:L'Hôpital's rule
285:Taylor's theorem
206:
205:
202:
146:
144:
143:
138:
90:
81:
76:
46:
45:
21:
17978:
17977:
17973:
17972:
17971:
17969:
17968:
17967:
17928:
17927:
17926:
17921:
17910:
17859:P-adic analysis
17810:
17796:Matrix calculus
17791:Tensor calculus
17786:Vector calculus
17749:Differentiation
17729:
17723:
17693:
17688:
17684:Steinmetz solid
17669:Integration Bee
17603:
17585:
17511:
17453:Colin Maclaurin
17429:
17397:
17391:
17263:
17257:Tensor calculus
17234:Volume integral
17170:
17145:Basic theorems
17108:Vector calculus
17102:
16983:
16950:Newton's method
16785:
16764:One-sided limit
16740:
16721:Rolle's theorem
16711:Linear function
16662:
16657:
16570:
16567:
16562:
16557:
16534:
16512:
16485:
16483:
16481:
16466:
16447:
16431:Strang, Gilbert
16424:
16399:
16381:
16353:
16329:
16309:
16281:
16255:
16237:
16215:
16195:
16153:
16147:Jarník, Vojtěch
16123:
16113:
16084:
16066:
16038:
16018:
15984:
15956:
15927:
15899:
15881:
15867:Evans, Lawrence
15860:
15824:
15766:
15748:Springer-Verlag
15721:
15703:
15668:
15630:10.2307/1967725
15614:Cajori, Florian
15607:
15514:
15486:
15468:Apostol, Tom M.
15462:
15457:
15456:
15448:
15444:
15432:
15428:
15412:
15408:
15396:
15392:
15376:
15372:
15360:
15356:
15348:
15344:
15336:
15332:
15320:
15316:
15308:
15304:
15292:
15283:
15271:
15267:
15259:
15255:
15247:
15243:
15231:
15227:
15207:
15203:
15191:
15187:
15171:
15167:
15159:
15152:
15144:
15140:
15128:
15124:
15112:
15108:
15100:
15096:
15088:, p. 160;
15084:
15080:
15063:
15059:
15038:
15035:
15034:
15032:
15014:
15011:
15010:
15008:
14990:
14987:
14986:
14984:
14979:
14975:
14967:
14960:
14952:, p. 172;
14948:
14944:
14936:
14932:
14920:
14916:
14900:
14896:
14884:
14880:
14872:, p. 110;
14868:
14864:
14852:
14848:
14823:
14820:
14819:
14817:
14794:
14791:
14790:
14756:
14738:
14735:
14734:
14732:
14713:
14710:
14709:
14690:
14687:
14686:
14665:
14657:
14655:
14653:
14650:
14649:
14630:
14627:
14626:
14599:
14596:
14595:
14593:
14589:
14581:
14572:
14564:
14560:
14552:
14548:
14540:
14536:
14524:
14520:
14512:
14508:
14492:
14488:
14476:, p. 114;
14472:, p. 156;
14468:
14464:
14452:, p. 113;
14448:, p. 149;
14444:
14440:
14428:, p. 114;
14424:, p. 156;
14420:
14416:
14404:
14400:
14392:
14388:
14380:
14376:
14364:
14360:
14352:
14348:
14332:
14328:
14312:
14308:
14296:
14292:
14280:, p. 127;
14276:, p. 160;
14272:
14268:
14252:, p. 160;
14248:
14244:
14239:
14231:Lie derivative
14202:
14144:weak derivative
14095:
14092:
14091:
14089:
14073:
14070:
14069:
14066:tangent bundles
14038:
14035:
14034:
14032:
14016:
14013:
14012:
13996:
13993:
13992:
13976:
13973:
13972:
13950:
13947:
13946:
13944:
13928:
13925:
13924:
13896:
13893:
13892:
13873:
13868:
13867:
13865:
13862:
13861:
13859:
13835:
13832:
13831:
13815:
13812:
13811:
13775:
13770:
13769:
13767:
13764:
13763:
13746:
13741:
13740:
13738:
13735:
13734:
13718:
13716:
13713:
13712:
13696:
13694:
13691:
13690:
13663:
13660:
13659:
13657:
13641:
13638:
13637:
13620:
13615:
13614:
13612:
13609:
13608:
13592:
13590:
13587:
13586:
13568:
13566:
13563:
13562:
13560:
13544:
13542:
13539:
13538:
13519:
13513:
13511:Generalizations
13486:
13473:
13469:
13465:
13458:
13454:
13450:
13448:
13444:
13443:
13433:
13432:
13428:
13417:
13406:
13404:
13401:
13400:
13384:
13382:
13379:
13378:
13362:
13359:
13358:
13355:Jacobian matrix
13328:
13324:
13309:
13305:
13296:
13292:
13281:
13278:
13277:
13275:
13259:
13256:
13255:
13237:
13235:
13232:
13231:
13229:
13213:
13210:
13209:
13193:
13185:
13174:
13172:
13169:
13168:
13150:
13148:
13145:
13144:
13142:
13124:
13122:
13119:
13118:
13116:
13100:
13097:
13096:
13078:
13076:
13073:
13072:
13070:
13051:
13048:
13047:
13031:
13029:
13026:
13025:
13005:
12997:
12986:
12984:
12981:
12980:
12962:
12959:
12958:
12956:
12940:
12937:
12936:
12919:
12914:
12913:
12911:
12908:
12907:
12888:
12883:
12882:
12880:
12877:
12876:
12874:
12873:is a vector in
12858:
12850:
12839:
12837:
12834:
12833:
12816:
12811:
12810:
12808:
12805:
12804:
12785:
12780:
12779:
12777:
12774:
12773:
12771:
12770:is a vector in
12755:
12753:
12750:
12749:
12721:
12717:
12705:
12697:
12686:
12675:
12655:
12647:
12637:
12635:
12622:
12621:
12615:
12612:
12611:
12594:
12589:
12588:
12579:
12574:
12573:
12562:
12551:
12549:
12546:
12545:
12529:
12527:
12524:
12523:
12507:
12504:
12503:
12484:
12473:
12471:
12468:
12467:
12448:
12440:
12429:
12418:
12401:
12393:
12385:
12382:
12381:
12363:
12361:
12358:
12357:
12355:
12339:
12337:
12334:
12333:
12317:
12314:
12313:
12289:
12286:
12285:
12283:
12267:
12264:
12263:
12247:
12244:
12243:
12224:
12219:
12218:
12216:
12213:
12212:
12210:
12193:
12188:
12187:
12185:
12182:
12181:
12165:
12162:
12161:
12158:
12152:
12125:
12121:
12117:
12109:
12107:
12101:
12097:
12091:
12080:
12065:
12057:
12050:
12049:
12045:
12043:
12040:
12039:
12023:
12021:
12018:
12017:
12001:
11998:
11997:
11979:
11977:
11974:
11973:
11971:
11955:
11952:
11951:
11921:
11904:
11893:
11886:
11884:
11872:
11857:
11849:
11842:
11841:
11837:
11835:
11832:
11831:
11815:
11813:
11810:
11809:
11793:
11791:
11788:
11787:
11771:
11768:
11767:
11741:
11737:
11722:
11718:
11707:
11705:
11702:
11701:
11699:
11675:
11672:
11671:
11669:
11653:
11650:
11649:
11633:
11630:
11629:
11613:
11610:
11609:
11593:
11590:
11589:
11571:
11568:
11567:
11565:
11549:
11546:
11545:
11529:
11526:
11525:
11509:
11506:
11505:
11486:
11481:
11480:
11478:
11475:
11474:
11472:
11456:
11453:
11452:
11449:
11443:
11414:
11410:
11395:
11391:
11380:
11377:
11376:
11356:
11352:
11337:
11333:
11328:
11325:
11324:
11305:
11302:
11301:
11282:
11279:
11278:
11262:
11259:
11258:
11242:
11239:
11238:
11206:
11202:
11187:
11183:
11171:
11167:
11163:
11155:
11153:
11135:
11131:
11116:
11112:
11100:
11096:
11092:
11084:
11082:
11081:
11077:
11065:
11061:
11046:
11042:
11031:
11028:
11027:
11005:
11001:
10986:
10982:
10977:
10974:
10973:
10971:
10954:
10950:
10948:
10945:
10944:
10928:
10925:
10924:
10900:
10896:
10881:
10877:
10869:
10866:
10865:
10830:
10826:
10811:
10807:
10792:
10788:
10770:
10766:
10745:
10741:
10726:
10722:
10715:
10713:
10701:
10685:
10681:
10666:
10662:
10650:
10646:
10642:
10634:
10632:
10630:
10627:
10626:
10606:
10602:
10587:
10583:
10578:
10575:
10574:
10557:
10553:
10551:
10548:
10547:
10527:
10523:
10508:
10504:
10496:
10493:
10492:
10453:
10445:
10443:
10414:
10406:
10404:
10402:
10399:
10398:
10382:
10379:
10378:
10362:
10359:
10358:
10342:
10339:
10338:
10319:
10315:
10297:
10293:
10270:
10267:
10266:
10264:
10232:
10229:
10228:
10225:
10204:
10196:
10194:
10192:
10189:
10188:
10164:
10159:
10157:
10154:
10153:
10133:
10129:
10127:
10124:
10123:
10103:
10097:
10094:
10093:
10076:
10072:
10070:
10067:
10066:
10048:
10045:
10044:
10007:
10004:
10003:
9996:vector calculus
9984:
9978:
9953:
9952:
9950:
9947:
9946:
9928:
9925:
9924:
9922:
9906:
9904:
9901:
9900:
9866:
9843:
9842:
9840:
9828:
9803:
9802:
9800:
9797:
9796:
9763:
9761:
9758:
9757:
9740:
9735:
9734:
9732:
9729:
9728:
9711:
9706:
9705:
9703:
9700:
9699:
9666:
9662:
9638:
9634:
9616:
9612:
9601:
9599:
9596:
9595:
9594:, meaning that
9569:
9565:
9541:
9537:
9519:
9515:
9513:
9510:
9509:
9492:
9487:
9486:
9484:
9481:
9480:
9461:
9459:
9456:
9455:
9449:
9444:
9438:Vector calculus
9434:
9369:
9365:
9363:
9360:
9359:
9357:
9336:
9333:
9332:
9331:
9313:
9310:
9309:
9292:
9289:
9288:
9263:
9259:
9257:
9254:
9253:
9251:
9235:
9232:
9231:
9228:discussed above
9208:
9205:
9204:
9202:
9170:
9167:
9166:
9164:
9146:
9143:
9142:
9140:
9119:
9117:
9114:
9113:
9111:
9088:
9086:
9083:
9082:
9061:
9059:
9056:
9055:
9053:
9030:
9028:
9025:
9024:
9003:
9001:
8998:
8997:
8995:
8979:
8976:
8975:
8959:
8956:
8955:
8949:
8928:
8925:
8924:
8907:
8903:
8883:
8880:
8879:
8851:
8848:
8847:
8819:
8816:
8815:
8798:
8794:
8792:
8789:
8788:
8771:
8767:
8765:
8762:
8761:
8739:
8738:
8729:
8725:
8701:
8697:
8687:
8674:
8670:
8666:
8645:
8641:
8629:
8628:
8612:
8601:
8597:
8593:
8589:
8587:
8563:
8559:
8549:
8538:
8531:
8527:
8523:
8521:
8508:
8504:
8500:
8484:
8473:
8469:
8465:
8461:
8459:
8438:
8434:
8424:
8407:
8403:
8401:
8398:
8397:
8374:
8370:
8342:
8338:
8334:
8319:
8315:
8298:
8295:
8294:
8291:
8253:
8224:
8204:
8202:
8199:
8198:
8145:
8142:
8141:
8139:
8118:
8102:
8099:
8098:
8082:
8079:
8078:
8059:
8055:
8046:
8029:
8028:
8026:
8005:
8001:
8000:
7998:
7995:
7994:
7942:
7940:
7937:
7936:
7920:
7917:
7916:
7897:
7883:
7872:
7869:
7868:
7850:
7847:
7846:
7844:
7828:
7825:
7824:
7805:
7788:
7777:
7766:
7763:
7762:
7732:
7729:
7728:
7726:
7710:
7707:
7706:
7690:
7687:
7686:
7670:
7667:
7666:
7647:
7633:
7619:
7599:
7596:
7595:
7550:
7548:
7545:
7544:
7524:
7521:
7520:
7518:
7502:
7499:
7498:
7475:
7472:
7471:
7455:
7452:
7451:
7450:Given that the
7448:
7421:
7417:
7410:
7405:
7377:
7372:
7370:
7367:
7366:
7334:
7331:
7330:
7310:
7306:
7294:
7263:
7258:
7256:
7253:
7252:
7220:
7217:
7216:
7196:
7192:
7180:
7152:
7147:
7145:
7142:
7141:
7101:
7097:
7067:
7063:
7062:
7057:
7036:
7032:
7004:
6999:
6997:
6994:
6993:
6935:
6930:
6928:
6925:
6924:
6869:
6864:
6862:
6859:
6858:
6819:
6816:
6815:
6782:
6777:
6756:
6752:
6742:
6737:
6735:
6732:
6731:
6708:
6705:
6704:
6683:
6655:
6650:
6648:
6645:
6644:
6621:
6618:
6617:
6585:
6581:
6572:
6568:
6558:
6553:
6551:
6548:
6547:
6529:
6525:
6516:
6512:
6502:
6497:
6495:
6492:
6491:
6447:
6443:
6431:
6427:
6417:
6412:
6410:
6407:
6406:
6390:
6372:
6369:
6368:
6352:
6349:
6348:
6345:
6337:differentiation
6328:
6322:
6297:
6296:
6268:
6263:
6257:
6256:
6246:
6241:
6214:
6209:
6202:
6199:
6198:
6196:
6179:
6178:
6150:
6145:
6139:
6138:
6128:
6123:
6093:
6089:
6087:
6084:
6083:
6046:
6042:
6040:
6037:
6036:
6034:
6014:
6010:
6008:
6005:
6004:
6003:can be written
5988:
5985:
5984:
5945:
5942:
5941:
5939:
5896:
5892:
5890:
5887:
5886:
5884:
5868:
5865:
5864:
5836:
5833:
5832:
5814:
5811:
5810:
5808:
5761:
5760:
5758:
5755:
5754:
5752:
5730:
5729:
5727:
5724:
5723:
5705:
5702:
5701:
5699:
5683:
5680:
5679:
5650:
5647:
5646:
5644:
5626:
5623:
5622:
5620:
5597:
5593:
5591:
5588:
5587:
5562:
5558:
5556:
5553:
5552:
5550:
5529:
5528:
5524:
5522:
5519:
5518:
5493:
5491:
5488:
5487:
5485:
5466:
5464:
5461:
5460:
5442:
5439:
5438:
5436:
5415:
5413:
5410:
5409:
5407:
5389:
5386:
5385:
5383:
5365:
5362:
5361:
5359:
5329:
5327:
5324:
5323:
5321:
5286:
5283:
5282:
5280:
5250:
5242:
5240:
5227:
5219:
5217:
5204:
5196:
5194:
5192:
5189:
5188:
5148:
5145:
5144:
5113:
5110:
5109:
5080:
5079:
5057:
5052:
5046:
5045:
5035:
5030:
5018:
5014:
5010:
5000:
4996:
4995:
4993:
4991:
4988:
4987:
4956:
4953:
4952:
4936:
4933:
4932:
4912:
4908:
4904:
4894:
4890:
4889:
4887:
4885:
4882:
4881:
4845:
4840:
4827:
4819:
4817:
4815:
4812:
4811:
4810:to a function,
4789:
4786:
4785:
4783:
4767:
4764:
4763:
4739:
4731:
4729:
4726:
4723:
4722:
4720:
4685:
4682:
4681:
4660:
4657:
4656:
4654:
4635:
4632:
4631:
4616:
4610:
4552:
4549:
4548:
4527:
4523:
4519:
4502:
4499:
4498:
4472:
4469:
4468:
4447:
4444:
4443:
4441:
4425:
4422:
4421:
4405:
4402:
4401:
4385:
4382:
4381:
4365:
4362:
4361:
4345:
4342:
4341:
4317:
4314:
4313:
4311:
4295:
4287:
4270:
4267:
4266:
4239:
4236:
4235:
4219:
4216:
4215:
4193:
4190:
4189:
4173:
4170:
4169:
4153:
4150:
4149:
4127:
4124:
4123:
4107:
4104:
4103:
4081:
4078:
4077:
4061:
4058:
4057:
4041:
4038:
4037:
4021:
4018:
4017:
4001:
3998:
3997:
3981:
3978:
3977:
3959:
3956:
3955:
3953:
3937:
3934:
3933:
3913:
3910:
3909:
3893:
3890:
3889:
3873:
3870:
3869:
3849:
3846:
3845:
3827:
3824:
3823:
3821:
3801:
3798:
3797:
3794:
3793:
3792:
3791:
3790:
3783:
3780:
3772:
3771:
3764:
3753:
3735:
3734:
3716:
3715:
3694:
3690:
3675:
3674:
3659:
3652:
3648:
3638:
3636:
3623:
3606:
3604:
3603:
3599:
3584:
3583:
3569:
3562:
3558:
3530:
3528:
3524:
3509:
3508:
3494:
3487:
3483:
3474:
3470:
3434:
3430:
3429:
3427:
3423:
3410:
3393:
3389:
3387:
3384:
3383:
3366:
3362:
3345:
3342:
3341:
3321:
3318:
3317:
3296:
3293:
3292:
3290:
3265:
3227:
3225:
3221:
3195:
3193:
3190:
3189:
3164:
3161:
3160:
3144:
3141:
3140:
3098:
3095:
3094:
3062:
3059:
3058:
3024:
3016:
3014:
3012:
3009:
3008:
3005:
2984:
2981:
2980:
2964:
2961:
2960:
2940:
2937:
2936:
2887:
2884:
2883:
2846:
2843:
2842:
2824:
2821:
2820:
2818:
2815:
2814:
2813:
2812:
2811:
2791:
2787:
2783:
2771:
2767:
2751:
2747:
2743:
2735:
2732:
2731:
2728:
2720:
2719:
2704:
2654:
2652:
2649:
2648:
2646:
2627:
2624:
2623:
2607:
2604:
2603:
2584:
2581:
2580:
2562:
2559:
2558:
2556:
2540:
2537:
2536:
2514:
2511:
2510:
2469:
2465:
2456:
2452:
2431:
2427:
2426:
2424:
2409:
2405:
2396:
2392:
2379:
2377:
2334:
2332:
2330:
2327:
2326:
2309:
2305:
2288:
2285:
2284:
2268:
2265:
2264:
2245:
2242:
2241:
2223:
2220:
2219:
2217:
2189:
2187:
2184:
2183:
2155:
2153:
2150:
2149:
2133:
2130:
2129:
2113:
2110:
2109:
2108:. The function
2091:
2088:
2087:
2085:
2059:
2057:
2054:
2053:
2037:
2034:
2033:
2017:
2014:
2013:
1997:
1994:
1993:
1973:
1970:
1969:
1965:§ Notation
1945:
1942:
1941:
1939:
1920:
1917:
1916:
1895:
1892:
1891:
1889:
1871:
1868:
1867:
1865:
1849:
1846:
1845:
1829:
1826:
1825:
1792:
1784:
1782:
1779:
1776:
1775:
1773:
1755:
1752:
1751:
1749:
1731:
1728:
1727:
1725:
1695:
1693:
1690:
1689:
1687:
1686:can be denoted
1671:
1668:
1667:
1651:
1648:
1647:
1631:
1628:
1627:
1611:
1608:
1607:
1587:
1584:
1583:
1565:
1562:
1561:
1559:
1543:
1540:
1539:
1463:
1461:
1454:
1450:
1448:
1445:
1444:
1413:
1410:
1409:
1387:
1384:
1383:
1361:
1353:
1351:
1348:
1347:
1331:
1328:
1327:
1311:
1308:
1307:
1289:
1286:
1285:
1283:
1229:
1227:
1215:
1203:
1200:
1199:
1178:
1175:
1174:
1172:
1148:
1145:
1144:
1115:
1112:
1111:
1105:
1100:
1092:gradient vector
1076:Jacobian matrix
1030:differentiation
990:
961:
960:
946:Integration Bee
921:
918:
911:
910:
886:
883:
876:
875:
848:
845:
838:
837:
819:Volume integral
754:
749:
742:
741:
658:
653:
646:
645:
615:
534:
529:
522:
521:
513:Risch algorithm
483:Euler's formula
369:
364:
357:
356:
338:General Leibniz
221:generalizations
203:
198:
191:
177:Rolle's theorem
172:
147:
83:
77:
72:
66:
63:
62:
42:
35:
28:
23:
22:
15:
12:
11:
5:
17976:
17966:
17965:
17960:
17955:
17950:
17945:
17940:
17923:
17922:
17915:
17912:
17911:
17909:
17908:
17903:
17898:
17893:
17888:
17883:
17877:
17876:
17871:
17869:Measure theory
17866:
17863:P-adic numbers
17856:
17851:
17846:
17841:
17836:
17826:
17821:
17815:
17812:
17811:
17809:
17808:
17803:
17798:
17793:
17788:
17783:
17778:
17773:
17772:
17771:
17766:
17761:
17751:
17746:
17734:
17731:
17730:
17722:
17721:
17714:
17707:
17699:
17690:
17689:
17687:
17686:
17681:
17676:
17671:
17666:
17664:Gabriel's horn
17661:
17656:
17655:
17654:
17649:
17644:
17639:
17634:
17626:
17625:
17624:
17615:
17613:
17609:
17608:
17605:
17604:
17602:
17601:
17596:
17594:List of limits
17590:
17587:
17586:
17584:
17583:
17582:
17581:
17576:
17571:
17561:
17560:
17559:
17549:
17544:
17539:
17534:
17528:
17526:
17517:
17513:
17512:
17510:
17509:
17502:
17495:
17493:Leonhard Euler
17490:
17485:
17480:
17475:
17470:
17465:
17460:
17455:
17450:
17445:
17439:
17437:
17431:
17430:
17428:
17427:
17422:
17417:
17412:
17407:
17401:
17399:
17393:
17392:
17390:
17389:
17388:
17387:
17382:
17377:
17372:
17367:
17362:
17357:
17352:
17347:
17342:
17334:
17333:
17332:
17327:
17326:
17325:
17320:
17310:
17305:
17300:
17295:
17290:
17285:
17277:
17271:
17269:
17265:
17264:
17262:
17261:
17260:
17259:
17254:
17249:
17244:
17236:
17231:
17226:
17221:
17216:
17211:
17206:
17201:
17196:
17194:Hessian matrix
17191:
17186:
17180:
17178:
17172:
17171:
17169:
17168:
17167:
17166:
17161:
17156:
17151:
17149:Line integrals
17143:
17142:
17141:
17136:
17131:
17126:
17121:
17112:
17110:
17104:
17103:
17101:
17100:
17095:
17090:
17089:
17088:
17083:
17075:
17070:
17069:
17068:
17058:
17057:
17056:
17051:
17046:
17036:
17031:
17030:
17029:
17019:
17014:
17009:
17004:
16999:
16997:Antiderivative
16993:
16991:
16985:
16984:
16982:
16981:
16980:
16979:
16974:
16969:
16959:
16958:
16957:
16952:
16944:
16943:
16942:
16937:
16932:
16927:
16917:
16916:
16915:
16910:
16905:
16900:
16892:
16891:
16890:
16885:
16884:
16883:
16873:
16868:
16863:
16858:
16853:
16843:
16842:
16841:
16836:
16826:
16821:
16816:
16811:
16806:
16801:
16795:
16793:
16787:
16786:
16784:
16783:
16778:
16773:
16768:
16767:
16766:
16756:
16750:
16748:
16742:
16741:
16739:
16738:
16733:
16728:
16723:
16718:
16713:
16708:
16703:
16698:
16693:
16688:
16683:
16678:
16672:
16670:
16664:
16663:
16656:
16655:
16648:
16641:
16633:
16625:
16624:
16614:
16595:
16586:
16566:
16565:External links
16563:
16561:
16560:
16555:
16537:
16533:978-0131469686
16532:
16515:
16510:
16492:
16479:
16450:
16445:
16427:
16422:
16406:Stewart, James
16402:
16397:
16384:
16379:
16364:
16356:
16351:
16332:
16327:
16312:
16307:
16284:
16279:
16258:
16253:
16240:
16235:
16218:
16213:
16198:
16193:
16178:
16164:(4): 248–264,
16143:
16116:
16111:
16087:
16082:
16069:
16064:
16041:
16036:
16021:
16016:
15998:
15987:
15982:
15959:
15954:
15930:
15925:
15902:
15897:
15884:
15879:
15863:
15858:
15835:
15822:
15801:
15769:
15764:
15732:
15719:
15706:
15701:
15680:
15671:
15666:
15651:
15610:
15605:
15592:
15572:(2): 117–122.
15555:
15543:(3): 174–179,
15529:Banach, Stefan
15525:
15512:
15489:
15484:
15463:
15461:
15458:
15455:
15454:
15442:
15426:
15406:
15390:
15370:
15354:
15342:
15340:, p. 303.
15330:
15314:
15302:
15281:
15265:
15263:, p. 642.
15253:
15241:
15225:
15209:Silverman 1989
15201:
15185:
15183:, p. 682.
15165:
15163:, p. 893.
15150:
15148:, p. 193.
15138:
15130:Carothers 2000
15122:
15106:
15094:
15078:
15057:
15042:
15018:
14994:
14973:
14971:, p. 160.
14958:
14942:
14930:
14924:, p. 63;
14914:
14912:, p. 171.
14894:
14892:, p. 228.
14878:
14862:
14846:
14827:
14816:of a function
14801:
14798:
14772:
14769:
14766:
14762:
14759:
14755:
14752:
14749:
14746:
14743:
14720:
14717:
14697:
14694:
14671:
14668:
14663:
14660:
14637:
14634:
14606:
14603:
14587:
14570:
14568:, p. 110.
14558:
14556:, p. 204.
14546:
14544:, p. 172.
14534:
14518:
14506:
14486:
14462:
14438:
14414:
14408:, p. 45;
14398:
14386:
14374:
14372:, p. 128.
14358:
14346:
14336:, p. 83;
14326:
14316:, p. 88;
14306:
14300:, p. 83;
14290:
14266:
14241:
14240:
14238:
14235:
14234:
14233:
14228:
14223:
14218:
14213:
14208:
14201:
14198:
14197:
14196:
14181:
14170:
14151:
14136:
14117:
14099:
14077:
14051:
14048:
14045:
14042:
14020:
14000:
13980:
13954:
13932:
13912:
13909:
13906:
13903:
13900:
13876:
13871:
13856:smooth surface
13839:
13819:
13804:
13793:complex linear
13778:
13773:
13749:
13744:
13721:
13699:
13676:
13673:
13670:
13667:
13645:
13623:
13618:
13595:
13571:
13547:
13515:Main article:
13512:
13509:
13497:
13492:
13489:
13484:
13476:
13472:
13468:
13461:
13457:
13453:
13447:
13442:
13436:
13431:
13427:
13424:
13420:
13416:
13412:
13409:
13387:
13366:
13336:
13331:
13327:
13323:
13320:
13317:
13312:
13308:
13304:
13299:
13295:
13291:
13288:
13285:
13263:
13240:
13217:
13196:
13192:
13188:
13184:
13180:
13177:
13153:
13141:, and for all
13127:
13104:
13081:
13055:
13034:
13020:is called the
13008:
13004:
13000:
12996:
12992:
12989:
12966:
12944:
12922:
12917:
12891:
12886:
12861:
12857:
12853:
12849:
12845:
12842:
12819:
12814:
12788:
12783:
12758:
12737:
12734:
12728:
12724:
12720:
12715:
12712:
12708:
12704:
12700:
12696:
12692:
12689:
12685:
12682:
12678:
12674:
12671:
12668:
12665:
12662:
12658:
12654:
12650:
12646:
12643:
12640:
12632:
12629:
12625:
12620:
12597:
12592:
12587:
12582:
12577:
12572:
12569:
12565:
12561:
12557:
12554:
12532:
12511:
12491:
12487:
12483:
12479:
12476:
12455:
12451:
12447:
12443:
12439:
12435:
12432:
12428:
12425:
12421:
12417:
12414:
12411:
12408:
12404:
12400:
12396:
12392:
12389:
12366:
12342:
12321:
12299:
12296:
12293:
12271:
12251:
12227:
12222:
12196:
12191:
12169:
12154:Main article:
12151:
12148:
12136:
12128:
12124:
12120:
12115:
12112:
12104:
12100:
12094:
12089:
12086:
12083:
12079:
12075:
12072:
12068:
12064:
12060:
12053:
12048:
12026:
12005:
11982:
11959:
11937:
11932:
11928:
11924:
11920:
11917:
11914:
11911:
11907:
11903:
11900:
11896:
11892:
11889:
11881:
11878:
11875:
11871:
11867:
11864:
11860:
11856:
11852:
11845:
11840:
11818:
11796:
11775:
11749:
11744:
11740:
11736:
11733:
11730:
11725:
11721:
11717:
11714:
11710:
11685:
11682:
11679:
11657:
11637:
11617:
11597:
11575:
11553:
11533:
11513:
11489:
11484:
11460:
11445:Main article:
11442:
11439:
11422:
11417:
11413:
11409:
11406:
11403:
11398:
11394:
11390:
11387:
11384:
11375:to the vector
11364:
11359:
11355:
11351:
11348:
11345:
11340:
11336:
11332:
11312:
11309:
11286:
11266:
11246:
11222:
11218:
11214:
11209:
11205:
11201:
11198:
11195:
11190:
11186:
11182:
11174:
11170:
11166:
11161:
11158:
11152:
11149:
11146:
11143:
11138:
11134:
11130:
11127:
11124:
11119:
11115:
11111:
11103:
11099:
11095:
11090:
11087:
11080:
11076:
11073:
11068:
11064:
11060:
11057:
11054:
11049:
11045:
11041:
11038:
11035:
11013:
11008:
11004:
11000:
10997:
10994:
10989:
10985:
10981:
10957:
10953:
10932:
10908:
10903:
10899:
10895:
10892:
10889:
10884:
10880:
10876:
10873:
10847:
10842:
10838:
10833:
10829:
10825:
10822:
10819:
10814:
10810:
10806:
10803:
10800:
10795:
10791:
10787:
10784:
10781:
10778:
10773:
10769:
10765:
10762:
10759:
10756:
10753:
10748:
10744:
10740:
10737:
10734:
10729:
10725:
10721:
10718:
10710:
10707:
10704:
10700:
10696:
10693:
10688:
10684:
10680:
10677:
10674:
10669:
10665:
10661:
10653:
10649:
10645:
10640:
10637:
10614:
10609:
10605:
10601:
10598:
10595:
10590:
10586:
10582:
10560:
10556:
10535:
10530:
10526:
10522:
10519:
10516:
10511:
10507:
10503:
10500:
10480:
10477:
10474:
10471:
10468:
10465:
10459:
10456:
10451:
10448:
10441:
10438:
10435:
10432:
10429:
10426:
10420:
10417:
10412:
10409:
10386:
10366:
10346:
10322:
10318:
10314:
10311:
10308:
10305:
10300:
10296:
10292:
10289:
10286:
10283:
10280:
10277:
10274:
10236:
10210:
10207:
10202:
10199:
10176:
10170:
10167:
10163:
10141:
10136:
10132:
10110:
10106:
10102:
10079:
10075:
10065:
10052:
10032:
10029:
10026:
10023:
10020:
10017:
10014:
10011:
9980:Main article:
9977:
9974:
9960:
9956:
9932:
9909:
9888:
9883:
9879:
9876:
9873:
9869:
9865:
9862:
9859:
9856:
9853:
9850:
9846:
9837:
9834:
9831:
9827:
9823:
9820:
9817:
9814:
9810:
9806:
9793:tangent vector
9776:
9773:
9770:
9766:
9743:
9738:
9714:
9709:
9683:
9680:
9677:
9674:
9669:
9665:
9661:
9658:
9655:
9652:
9649:
9646:
9641:
9637:
9633:
9630:
9627:
9624:
9619:
9615:
9611:
9608:
9604:
9583:
9580:
9577:
9572:
9568:
9564:
9561:
9558:
9555:
9552:
9549:
9544:
9540:
9536:
9533:
9530:
9527:
9522:
9518:
9495:
9490:
9464:
9448:
9445:
9433:
9430:
9372:
9368:
9340:
9317:
9296:
9272:
9269:
9266:
9262:
9239:
9226:. As has been
9212:
9186:
9183:
9180:
9177:
9174:
9150:
9125:
9122:
9094:
9091:
9067:
9064:
9036:
9033:
9009:
9006:
8983:
8963:
8948:
8945:
8932:
8910:
8906:
8902:
8899:
8896:
8893:
8890:
8887:
8867:
8864:
8861:
8858:
8855:
8835:
8832:
8829:
8826:
8823:
8801:
8797:
8774:
8770:
8737:
8732:
8728:
8724:
8721:
8718:
8715:
8712:
8709:
8704:
8700:
8694:
8691:
8686:
8682:
8677:
8673:
8669:
8665:
8662:
8659:
8656:
8653:
8648:
8644:
8640:
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8634:
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8630:
8627:
8624:
8618:
8615:
8609:
8604:
8600:
8596:
8592:
8586:
8583:
8580:
8577:
8574:
8571:
8566:
8562:
8555:
8552:
8546:
8541:
8537:
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8526:
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8516:
8511:
8507:
8503:
8499:
8496:
8490:
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8476:
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8468:
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8458:
8453:
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8447:
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8441:
8437:
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8427:
8425:
8423:
8420:
8417:
8413:
8410:
8406:
8405:
8385:
8382:
8377:
8373:
8369:
8366:
8363:
8360:
8357:
8354:
8350:
8345:
8341:
8337:
8333:
8330:
8327:
8322:
8318:
8314:
8311:
8308:
8305:
8302:
8290:
8287:
8286:
8285:
8284:
8283:
8272:
8269:
8266:
8263:
8259:
8256:
8252:
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8234:
8230:
8227:
8223:
8220:
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8210:
8207:
8182:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8152:
8149:
8127:
8126:
8125:
8106:
8086:
8062:
8058:
8052:
8049:
8045:
8042:
8039:
8035:
8032:
8025:
8021:
8017:
8012:
8009:
8004:
7984:
7983:
7982:
7970:
7967:
7964:
7961:
7958:
7955:
7952:
7948:
7945:
7924:
7903:
7900:
7896:
7893:
7889:
7886:
7882:
7879:
7876:
7854:
7832:
7811:
7808:
7804:
7801:
7798:
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7791:
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7783:
7780:
7776:
7773:
7770:
7752:
7751:
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7736:
7714:
7694:
7674:
7653:
7650:
7646:
7643:
7639:
7636:
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7629:
7625:
7622:
7618:
7615:
7612:
7609:
7606:
7603:
7585:
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7583:
7572:
7569:
7566:
7563:
7560:
7556:
7553:
7528:
7506:
7479:
7459:
7447:
7444:
7443:
7442:
7441:
7440:
7424:
7420:
7416:
7413:
7409:
7404:
7401:
7398:
7395:
7392:
7389:
7383:
7380:
7376:
7364:
7353:
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7341:
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7313:
7309:
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7131:
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7118:
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7112:
7109:
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7056:
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7050:
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7044:
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7028:
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6991:
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6887:
6884:
6881:
6875:
6872:
6868:
6848:
6847:
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6835:
6832:
6829:
6826:
6823:
6800:
6797:
6794:
6791:
6788:
6785:
6781:
6776:
6773:
6770:
6767:
6764:
6759:
6755:
6748:
6745:
6741:
6729:
6718:
6715:
6712:
6690:
6687:
6682:
6679:
6676:
6673:
6670:
6667:
6661:
6658:
6654:
6642:
6631:
6628:
6625:
6605:
6602:
6599:
6596:
6593:
6588:
6584:
6580:
6575:
6571:
6564:
6561:
6557:
6545:
6532:
6528:
6524:
6519:
6515:
6508:
6505:
6501:
6471:
6470:
6469:
6456:
6453:
6450:
6446:
6442:
6439:
6434:
6430:
6423:
6420:
6416:
6376:
6356:
6344:
6341:
6324:Main article:
6321:
6318:
6300:
6295:
6292:
6289:
6286:
6283:
6280:
6274:
6271:
6267:
6260:
6252:
6249:
6245:
6240:
6237:
6234:
6231:
6228:
6225:
6222:
6217:
6212:
6208:
6182:
6177:
6174:
6171:
6168:
6165:
6162:
6156:
6153:
6149:
6142:
6134:
6131:
6127:
6122:
6119:
6116:
6113:
6110:
6107:
6104:
6099:
6096:
6092:
6069:
6066:
6063:
6060:
6057:
6054:
6049:
6045:
6022:
6017:
6013:
5992:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5932:Leonhard Euler
5928:Euler notation
5913:
5910:
5907:
5904:
5899:
5895:
5872:
5852:
5849:
5846:
5843:
5840:
5818:
5768:
5765:
5737:
5734:
5709:
5687:
5654:
5630:
5606:
5603:
5600:
5596:
5571:
5568:
5565:
5561:
5535:
5532:
5527:
5499:
5496:
5472:
5469:
5446:
5421:
5418:
5393:
5369:
5345:
5342:
5339:
5335:
5332:
5314:prime notation
5299:
5296:
5293:
5290:
5262:
5256:
5253:
5248:
5245:
5239:
5233:
5230:
5225:
5222:
5216:
5210:
5207:
5202:
5199:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5132:
5129:
5126:
5123:
5120:
5117:
5088:
5083:
5078:
5075:
5072:
5069:
5063:
5060:
5056:
5049:
5041:
5038:
5034:
5029:
5021:
5017:
5013:
5008:
5003:
4999:
4975:
4972:
4969:
4966:
4963:
4960:
4940:
4915:
4911:
4907:
4902:
4897:
4893:
4869:
4866:
4863:
4860:
4857:
4851:
4848:
4844:
4839:
4833:
4830:
4825:
4822:
4793:
4771:
4745:
4742:
4737:
4734:
4704:
4701:
4698:
4695:
4692:
4689:
4667:
4664:
4642:
4639:
4612:Main article:
4609:
4606:
4562:
4559:
4556:
4534:
4530:
4526:
4522:
4518:
4515:
4512:
4509:
4506:
4482:
4479:
4476:
4454:
4451:
4429:
4409:
4389:
4369:
4349:
4327:
4324:
4321:
4298:
4294:
4290:
4286:
4283:
4280:
4277:
4274:
4263:absolute value
4249:
4246:
4243:
4223:
4203:
4200:
4197:
4177:
4157:
4137:
4134:
4131:
4111:
4091:
4088:
4085:
4065:
4045:
4025:
4005:
3985:
3963:
3941:
3917:
3897:
3877:
3853:
3831:
3818:differentiable
3805:
3781:
3774:
3773:
3765:
3758:
3757:
3756:
3755:
3754:
3752:
3749:
3733:
3730:
3727:
3724:
3721:
3719:
3717:
3713:
3709:
3706:
3703:
3700:
3697:
3693:
3689:
3686:
3683:
3680:
3678:
3676:
3672:
3665:
3662:
3655:
3651:
3647:
3644:
3641:
3635:
3629:
3626:
3621:
3618:
3615:
3612:
3609:
3602:
3598:
3595:
3592:
3589:
3587:
3585:
3581:
3575:
3572:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3527:
3523:
3520:
3517:
3514:
3512:
3510:
3506:
3500:
3497:
3490:
3486:
3482:
3477:
3473:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3437:
3433:
3426:
3422:
3419:
3416:
3413:
3411:
3409:
3406:
3403:
3399:
3396:
3392:
3391:
3369:
3365:
3361:
3358:
3355:
3352:
3349:
3325:
3303:
3300:
3277:
3271:
3268:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3224:
3220:
3217:
3214:
3211:
3208:
3205:
3201:
3198:
3177:
3174:
3171:
3168:
3148:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3066:
3042:
3039:
3036:
3030:
3027:
3022:
3019:
3004:
3001:
2988:
2968:
2944:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2828:
2799:
2794:
2790:
2786:
2782:
2779:
2774:
2770:
2766:
2763:
2759:
2754:
2750:
2746:
2742:
2739:
2729:
2722:
2721:
2705:
2698:
2697:
2696:
2695:
2694:
2679:
2676:
2673:
2670:
2667:
2664:
2660:
2657:
2634:
2631:
2611:
2591:
2588:
2566:
2544:
2524:
2521:
2518:
2498:
2495:
2492:
2489:
2486:
2483:
2478:
2472:
2468:
2464:
2459:
2455:
2451:
2448:
2445:
2442:
2439:
2434:
2430:
2423:
2418:
2412:
2408:
2404:
2399:
2395:
2391:
2388:
2385:
2382:
2376:
2371:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2312:
2308:
2304:
2301:
2298:
2295:
2292:
2272:
2249:
2227:
2205:
2202:
2199:
2195:
2192:
2171:
2168:
2165:
2161:
2158:
2137:
2117:
2095:
2065:
2062:
2041:
2021:
2001:
1977:
1949:
1927:
1924:
1902:
1899:
1875:
1853:
1833:
1810:
1807:
1804:
1798:
1795:
1790:
1787:
1759:
1735:
1711:
1708:
1705:
1701:
1698:
1675:
1655:
1635:
1615:
1591:
1569:
1547:
1529:absolute value
1516:
1513:
1510:
1506:
1500:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1460:
1457:
1453:
1432:
1429:
1426:
1423:
1420:
1417:
1397:
1394:
1391:
1371:
1368:
1364:
1360:
1356:
1335:
1315:
1293:
1266:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1224:
1221:
1218:
1214:
1210:
1207:
1182:
1152:
1141:differentiable
1128:
1125:
1122:
1119:
1104:
1101:
1099:
1096:
992:
991:
989:
988:
981:
974:
966:
963:
962:
959:
958:
953:
948:
943:
941:List of topics
938:
933:
928:
922:
917:
916:
913:
912:
909:
908:
903:
898:
893:
887:
882:
881:
878:
877:
872:
871:
870:
869:
864:
859:
849:
844:
843:
840:
839:
834:
833:
832:
831:
826:
821:
816:
811:
806:
801:
793:
792:
788:
787:
786:
785:
780:
775:
770:
762:
761:
755:
748:
747:
744:
743:
738:
737:
736:
735:
730:
725:
720:
715:
710:
702:
701:
697:
696:
695:
694:
689:
684:
679:
674:
669:
659:
652:
651:
648:
647:
642:
641:
640:
639:
634:
629:
624:
619:
613:
608:
603:
598:
593:
585:
584:
578:
577:
576:
575:
570:
565:
560:
555:
550:
535:
528:
527:
524:
523:
518:
517:
516:
515:
510:
505:
500:
498:Changing order
495:
485:
480:
462:
457:
452:
444:
443:
442:Integration by
439:
438:
437:
436:
431:
426:
421:
416:
406:
404:Antiderivative
398:
397:
393:
392:
391:
390:
385:
380:
370:
363:
362:
359:
358:
353:
352:
351:
350:
345:
340:
335:
330:
325:
320:
315:
310:
305:
297:
296:
290:
289:
288:
287:
282:
277:
272:
267:
262:
254:
253:
249:
248:
247:
246:
245:
244:
239:
234:
224:
211:
210:
204:
197:
196:
193:
192:
190:
189:
184:
179:
173:
171:
170:
165:
159:
158:
157:
149:
148:
136:
133:
130:
127:
124:
121:
118:
115:
112:
109:
106:
103:
99:
96:
93:
89:
86:
80:
75:
71:
61:
58:
57:
51:
50:
26:
9:
6:
4:
3:
2:
17975:
17964:
17961:
17959:
17956:
17954:
17951:
17949:
17946:
17944:
17941:
17939:
17936:
17935:
17933:
17920:
17919:
17913:
17907:
17904:
17902:
17899:
17897:
17894:
17892:
17889:
17887:
17884:
17882:
17879:
17878:
17875:
17872:
17870:
17867:
17864:
17860:
17857:
17855:
17852:
17850:
17847:
17845:
17842:
17840:
17837:
17834:
17830:
17827:
17825:
17822:
17820:
17819:Real analysis
17817:
17816:
17813:
17807:
17804:
17802:
17799:
17797:
17794:
17792:
17789:
17787:
17784:
17782:
17779:
17777:
17774:
17770:
17767:
17765:
17762:
17760:
17757:
17756:
17755:
17752:
17750:
17747:
17745:
17741:
17740:
17736:
17735:
17732:
17728:
17720:
17715:
17713:
17708:
17706:
17701:
17700:
17697:
17685:
17682:
17680:
17677:
17675:
17672:
17670:
17667:
17665:
17662:
17660:
17657:
17653:
17650:
17648:
17645:
17643:
17640:
17638:
17635:
17633:
17630:
17629:
17627:
17623:
17620:
17619:
17617:
17616:
17614:
17610:
17600:
17597:
17595:
17592:
17591:
17588:
17580:
17577:
17575:
17572:
17570:
17567:
17566:
17565:
17562:
17558:
17555:
17554:
17553:
17550:
17548:
17545:
17543:
17540:
17538:
17535:
17533:
17530:
17529:
17527:
17525:
17521:
17518:
17514:
17508:
17507:
17503:
17501:
17500:
17496:
17494:
17491:
17489:
17486:
17484:
17481:
17479:
17476:
17474:
17471:
17469:
17468:Infinitesimal
17466:
17464:
17461:
17459:
17456:
17454:
17451:
17449:
17446:
17444:
17441:
17440:
17438:
17436:
17432:
17426:
17423:
17421:
17418:
17416:
17413:
17411:
17408:
17406:
17403:
17402:
17400:
17394:
17386:
17383:
17381:
17378:
17376:
17373:
17371:
17368:
17366:
17363:
17361:
17358:
17356:
17353:
17351:
17348:
17346:
17343:
17341:
17338:
17337:
17335:
17331:
17328:
17324:
17321:
17319:
17316:
17315:
17314:
17311:
17309:
17306:
17304:
17301:
17299:
17296:
17294:
17291:
17289:
17286:
17284:
17281:
17280:
17278:
17276:
17273:
17272:
17270:
17266:
17258:
17255:
17253:
17250:
17248:
17245:
17243:
17240:
17239:
17237:
17235:
17232:
17230:
17227:
17225:
17222:
17220:
17217:
17215:
17212:
17210:
17209:Line integral
17207:
17205:
17202:
17200:
17197:
17195:
17192:
17190:
17187:
17185:
17182:
17181:
17179:
17177:
17173:
17165:
17162:
17160:
17157:
17155:
17152:
17150:
17147:
17146:
17144:
17140:
17137:
17135:
17132:
17130:
17127:
17125:
17122:
17120:
17117:
17116:
17114:
17113:
17111:
17109:
17105:
17099:
17096:
17094:
17091:
17087:
17084:
17082:
17081:Washer method
17079:
17078:
17076:
17074:
17071:
17067:
17064:
17063:
17062:
17059:
17055:
17052:
17050:
17047:
17045:
17044:trigonometric
17042:
17041:
17040:
17037:
17035:
17032:
17028:
17025:
17024:
17023:
17020:
17018:
17015:
17013:
17010:
17008:
17005:
17003:
17000:
16998:
16995:
16994:
16992:
16990:
16986:
16978:
16975:
16973:
16970:
16968:
16965:
16964:
16963:
16960:
16956:
16953:
16951:
16948:
16947:
16945:
16941:
16938:
16936:
16933:
16931:
16928:
16926:
16923:
16922:
16921:
16918:
16914:
16913:Related rates
16911:
16909:
16906:
16904:
16901:
16899:
16896:
16895:
16893:
16889:
16886:
16882:
16879:
16878:
16877:
16874:
16872:
16869:
16867:
16864:
16862:
16859:
16857:
16854:
16852:
16849:
16848:
16847:
16844:
16840:
16837:
16835:
16832:
16831:
16830:
16827:
16825:
16822:
16820:
16817:
16815:
16812:
16810:
16807:
16805:
16802:
16800:
16797:
16796:
16794:
16792:
16788:
16782:
16779:
16777:
16774:
16772:
16769:
16765:
16762:
16761:
16760:
16757:
16755:
16752:
16751:
16749:
16747:
16743:
16737:
16734:
16732:
16729:
16727:
16724:
16722:
16719:
16717:
16714:
16712:
16709:
16707:
16704:
16702:
16699:
16697:
16694:
16692:
16689:
16687:
16684:
16682:
16679:
16677:
16674:
16673:
16671:
16669:
16665:
16661:
16654:
16649:
16647:
16642:
16640:
16635:
16634:
16631:
16627:
16622:
16621:Wolfram Alpha
16618:
16615:
16610:
16609:
16604:
16601:
16596:
16594:
16590:
16587:
16583:
16579:
16578:
16573:
16569:
16568:
16558:
16552:
16548:
16547:
16542:
16538:
16535:
16529:
16525:
16521:
16516:
16513:
16507:
16503:
16502:
16497:
16493:
16482:
16480:0-321-87896-5
16476:
16472:
16465:
16464:
16459:
16455:
16451:
16448:
16442:
16438:
16437:
16432:
16428:
16425:
16419:
16414:
16413:
16407:
16403:
16400:
16398:9780486660974
16394:
16390:
16385:
16382:
16380:9781614445012
16376:
16372:
16371:
16365:
16362:
16357:
16354:
16348:
16344:
16340:
16339:
16333:
16330:
16324:
16320:
16319:
16313:
16310:
16304:
16300:
16296:
16292:
16291:
16285:
16282:
16276:
16272:
16268:
16264:
16259:
16256:
16250:
16246:
16241:
16238:
16236:0-486-66721-9
16232:
16228:
16224:
16219:
16216:
16210:
16206:
16205:
16199:
16196:
16190:
16186:
16185:
16179:
16176:
16171:
16167:
16163:
16159:
16152:
16148:
16144:
16141:
16137:
16133:
16129:
16122:
16117:
16114:
16108:
16104:
16100:
16096:
16092:
16091:Hewitt, Edwin
16088:
16085:
16079:
16075:
16070:
16067:
16061:
16057:
16053:
16049:
16048:
16042:
16039:
16033:
16030:, CRC Press,
16029:
16028:
16022:
16019:
16013:
16009:
16008:
16003:
16002:Gonick, Larry
15999:
15995:
15994:
15988:
15985:
15979:
15975:
15971:
15967:
15966:
15960:
15957:
15951:
15947:
15943:
15939:
15935:
15931:
15928:
15922:
15918:
15914:
15910:
15909:
15903:
15900:
15894:
15890:
15885:
15882:
15880:0-8218-0772-2
15876:
15872:
15868:
15864:
15861:
15855:
15851:
15847:
15843:
15842:
15836:
15833:
15829:
15825:
15819:
15815:
15811:
15807:
15802:
15798:
15793:
15788:
15783:
15779:
15775:
15770:
15767:
15761:
15757:
15753:
15749:
15745:
15741:
15737:
15733:
15730:
15726:
15722:
15720:9780321781079
15716:
15712:
15707:
15704:
15698:
15694:
15690:
15686:
15681:
15677:
15676:Real Analysis
15672:
15669:
15663:
15659:
15658:
15652:
15649:
15645:
15640:
15635:
15631:
15627:
15623:
15619:
15615:
15611:
15608:
15606:9788174464507
15602:
15598:
15593:
15589:
15585:
15580:
15575:
15571:
15567:
15566:
15561:
15556:
15551:
15546:
15542:
15538:
15534:
15530:
15526:
15523:
15519:
15515:
15509:
15505:
15501:
15497:
15496:
15490:
15487:
15481:
15476:
15475:
15470:(June 1967),
15469:
15465:
15464:
15451:
15446:
15439:
15435:
15434:Georgiev 2018
15430:
15423:
15419:
15415:
15410:
15403:
15399:
15394:
15387:
15383:
15379:
15374:
15367:
15363:
15358:
15351:
15346:
15339:
15334:
15327:
15323:
15318:
15312:, p. 72.
15311:
15306:
15299:
15295:
15290:
15288:
15286:
15278:
15274:
15269:
15262:
15257:
15250:
15245:
15238:
15234:
15229:
15222:
15218:
15217:Bhardwaj 2005
15214:
15210:
15205:
15198:
15194:
15189:
15182:
15178:
15174:
15169:
15162:
15157:
15155:
15147:
15142:
15135:
15131:
15126:
15119:
15115:
15110:
15103:
15098:
15091:
15087:
15082:
15075:
15071:
15067:
15061:
15040:
15016:
14992:
14982:
14977:
14970:
14965:
14963:
14955:
14951:
14946:
14939:
14934:
14927:
14926:Kreyszig 1991
14923:
14918:
14911:
14907:
14903:
14898:
14891:
14887:
14882:
14875:
14871:
14866:
14859:
14855:
14850:
14843:
14825:
14815:
14799:
14796:
14789:
14767:
14760:
14757:
14753:
14750:
14747:
14744:
14741:
14718:
14715:
14695:
14692:
14669:
14666:
14661:
14658:
14635:
14632:
14624:
14621:, p. 119 and
14620:
14604:
14601:
14591:
14584:
14579:
14577:
14575:
14567:
14562:
14555:
14550:
14543:
14538:
14531:
14527:
14522:
14515:
14510:
14503:
14499:
14495:
14490:
14483:
14479:
14475:
14471:
14466:
14459:
14455:
14451:
14447:
14442:
14435:
14431:
14427:
14423:
14418:
14412:, p. 66.
14411:
14407:
14402:
14395:
14390:
14383:
14382:Thompson 1998
14378:
14371:
14367:
14366:Thompson 1998
14362:
14355:
14350:
14343:
14339:
14335:
14330:
14323:
14319:
14315:
14310:
14304:, p. 60.
14303:
14299:
14294:
14287:
14283:
14279:
14275:
14270:
14263:
14259:
14255:
14251:
14246:
14242:
14232:
14229:
14227:
14224:
14222:
14219:
14217:
14214:
14212:
14209:
14207:
14204:
14203:
14194:
14190:
14186:
14182:
14179:
14175:
14171:
14168:
14164:
14160:
14156:
14152:
14149:
14148:distributions
14145:
14141:
14137:
14134:
14130:
14126:
14122:
14118:
14115:
14097:
14075:
14067:
14046:
14040:
14018:
13998:
13978:
13970:
13952:
13930:
13910:
13904:
13901:
13898:
13874:
13857:
13853:
13852:tangent space
13837:
13817:
13809:
13805:
13802:
13798:
13794:
13776:
13747:
13674:
13671:
13668:
13665:
13643:
13621:
13536:
13532:
13528:
13527:
13526:
13524:
13518:
13508:
13495:
13490:
13487:
13482:
13474:
13470:
13459:
13455:
13445:
13440:
13429:
13425:
13410:
13407:
13364:
13356:
13352:
13329:
13325:
13321:
13318:
13315:
13310:
13306:
13302:
13297:
13293:
13286:
13283:
13261:
13215:
13178:
13175:
13102:
13067:
13053:
13023:
12990:
12987:
12964:
12942:
12920:
12889:
12843:
12840:
12817:
12786:
12735:
12732:
12690:
12687:
12683:
12669:
12663:
12652:
12641:
12630:
12595:
12580:
12570:
12555:
12552:
12509:
12477:
12474:
12453:
12433:
12430:
12426:
12412:
12409:
12398:
12387:
12319:
12297:
12294:
12291:
12269:
12249:
12225:
12194:
12167:
12157:
12147:
12134:
12126:
12122:
12113:
12102:
12098:
12092:
12087:
12084:
12081:
12077:
12073:
12058:
12046:
12003:
11957:
11948:
11935:
11930:
11915:
11912:
11901:
11898:
11887:
11879:
11873:
11865:
11850:
11838:
11808:at the point
11773:
11765:
11742:
11738:
11734:
11731:
11728:
11723:
11719:
11712:
11683:
11680:
11677:
11655:
11635:
11615:
11595:
11573:
11551:
11531:
11511:
11487:
11458:
11448:
11438:
11436:
11415:
11411:
11407:
11404:
11401:
11396:
11392:
11385:
11357:
11353:
11349:
11346:
11343:
11338:
11334:
11310:
11300:
11284:
11264:
11244:
11236:
11220:
11216:
11207:
11203:
11199:
11196:
11193:
11188:
11184:
11172:
11168:
11159:
11150:
11147:
11144:
11136:
11132:
11128:
11125:
11122:
11117:
11113:
11101:
11097:
11088:
11078:
11074:
11066:
11062:
11058:
11055:
11052:
11047:
11043:
11036:
11006:
11002:
10998:
10995:
10992:
10987:
10983:
10955:
10951:
10930:
10922:
10901:
10897:
10893:
10890:
10887:
10882:
10878:
10871:
10863:
10858:
10845:
10840:
10831:
10827:
10823:
10820:
10817:
10812:
10808:
10804:
10801:
10798:
10793:
10789:
10782:
10779:
10771:
10767:
10763:
10760:
10757:
10754:
10751:
10746:
10742:
10738:
10735:
10732:
10727:
10723:
10716:
10708:
10702:
10694:
10686:
10682:
10678:
10675:
10672:
10667:
10663:
10651:
10647:
10638:
10607:
10603:
10599:
10596:
10593:
10588:
10584:
10573:at the point
10558:
10554:
10528:
10524:
10520:
10517:
10514:
10509:
10505:
10498:
10478:
10475:
10472:
10469:
10466:
10463:
10457:
10449:
10439:
10436:
10433:
10430:
10427:
10424:
10418:
10410:
10384:
10364:
10344:
10320:
10316:
10312:
10309:
10306:
10303:
10298:
10294:
10290:
10284:
10281:
10278:
10272:
10262:
10258:
10254:
10251:is a rounded
10250:
10234:
10208:
10200:
10174:
10168:
10139:
10134:
10108:
10104:
10100:
10077:
10073:
10064:
10050:
10027:
10024:
10021:
10018:
10015:
10009:
10001:
9997:
9993:
9989:
9983:
9973:
9958:
9930:
9886:
9881:
9874:
9863:
9857:
9854:
9851:
9835:
9829:
9821:
9815:
9808:
9794:
9791:, called the
9790:
9771:
9741:
9712:
9697:
9675:
9667:
9663:
9659:
9656:
9653:
9647:
9639:
9635:
9631:
9625:
9617:
9613:
9606:
9578:
9570:
9566:
9562:
9559:
9556:
9550:
9542:
9538:
9534:
9528:
9520:
9516:
9493:
9479:
9454:
9443:
9439:
9429:
9427:
9423:
9419:
9415:
9411:
9406:
9404:
9400:
9396:
9395:
9390:
9370:
9366:
9356:
9338:
9329:
9315:
9294:
9267:
9260:
9237:
9229:
9225:
9210:
9181:
9178:
9175:
9148:
9123:
9120:
9110:, denoted as
9109:
9092:
9089:
9065:
9062:
9052:, denoted as
9051:
9034:
9031:
9007:
9004:
8981:
8961:
8953:
8944:
8930:
8908:
8904:
8900:
8894:
8888:
8885:
8862:
8856:
8853:
8830:
8824:
8821:
8799:
8795:
8772:
8768:
8759:
8755:
8735:
8730:
8726:
8719:
8713:
8710:
8707:
8702:
8698:
8692:
8689:
8684:
8680:
8675:
8671:
8667:
8663:
8660:
8657:
8654:
8651:
8646:
8642:
8638:
8635:
8633:
8625:
8622:
8616:
8613:
8607:
8602:
8598:
8594:
8590:
8581:
8575:
8572:
8569:
8564:
8560:
8553:
8550:
8544:
8539:
8535:
8532:
8528:
8524:
8518:
8514:
8509:
8505:
8501:
8497:
8494:
8488:
8485:
8479:
8474:
8470:
8466:
8462:
8456:
8448:
8445:
8442:
8435:
8431:
8428:
8426:
8418:
8411:
8408:
8383:
8380:
8375:
8371:
8364:
8358:
8355:
8352:
8348:
8343:
8339:
8335:
8331:
8328:
8325:
8320:
8316:
8312:
8306:
8300:
8270:
8264:
8257:
8254:
8250:
8241:
8235:
8228:
8225:
8221:
8215:
8208:
8205:
8197:
8196:
8174:
8168:
8162:
8159:
8153:
8147:
8137:
8133:
8132:
8128:
8121:
8104:
8084:
8060:
8056:
8050:
8047:
8043:
8040:
8037:
8033:
8030:
8023:
8019:
8015:
8010:
8007:
8002:
7993:
7992:
7990:
7989:
7988:Quotient rule
7985:
7968:
7965:
7962:
7959:
7956:
7953:
7950:
7946:
7943:
7922:
7901:
7898:
7894:
7891:
7887:
7880:
7877:
7852:
7830:
7809:
7806:
7802:
7799:
7796:
7792:
7789:
7785:
7781:
7774:
7771:
7761:
7760:
7758:
7757:
7753:
7734:
7712:
7692:
7672:
7651:
7648:
7644:
7641:
7637:
7634:
7630:
7627:
7623:
7616:
7613:
7610:
7607:
7604:
7594:
7593:
7591:
7590:
7586:
7570:
7567:
7561:
7554:
7551:
7543:
7542:
7526:
7504:
7496:
7495:Constant rule
7493:
7492:
7491:
7477:
7457:
7422:
7418:
7414:
7411:
7407:
7402:
7396:
7390:
7387:
7381:
7378:
7374:
7365:
7351:
7348:
7345:
7342:
7339:
7336:
7311:
7307:
7303:
7300:
7296:
7291:
7288:
7282:
7276:
7273:
7267:
7264:
7260:
7251:
7237:
7234:
7231:
7228:
7225:
7222:
7197:
7193:
7189:
7186:
7182:
7177:
7171:
7165:
7162:
7156:
7153:
7149:
7140:
7139:
7137:
7136:
7132:
7113:
7107:
7102:
7098:
7094:
7091:
7088:
7079:
7073:
7068:
7064:
7059:
7054:
7048:
7042:
7037:
7033:
7029:
7023:
7017:
7014:
7008:
7005:
7001:
6992:
6975:
6969:
6966:
6963:
6960:
6954:
6948:
6945:
6939:
6936:
6932:
6923:
6906:
6900:
6897:
6894:
6888:
6882:
6879:
6873:
6870:
6866:
6857:
6856:
6854:
6853:
6849:
6833:
6830:
6827:
6824:
6821:
6795:
6789:
6786:
6783:
6779:
6774:
6768:
6762:
6757:
6753:
6746:
6743:
6739:
6730:
6716:
6713:
6710:
6688:
6685:
6680:
6674:
6668:
6665:
6659:
6656:
6652:
6643:
6629:
6626:
6623:
6600:
6594:
6591:
6586:
6582:
6578:
6573:
6569:
6562:
6559:
6555:
6546:
6530:
6526:
6522:
6517:
6513:
6506:
6503:
6499:
6490:
6489:
6487:
6485:
6481:
6477:
6474:Functions of
6472:
6454:
6451:
6448:
6444:
6440:
6437:
6432:
6428:
6421:
6418:
6414:
6405:
6404:
6402:
6401:
6397:
6396:
6395:
6393:
6374:
6354:
6340:
6338:
6334:
6327:
6317:
6290:
6287:
6284:
6278:
6272:
6250:
6238:
6232:
6229:
6226:
6220:
6215:
6210:
6206:
6172:
6169:
6166:
6160:
6154:
6132:
6120:
6114:
6111:
6108:
6102:
6097:
6094:
6090:
6064:
6061:
6058:
6052:
6047:
6043:
6020:
6015:
6011:
5990:
5965:
5962:
5959:
5953:
5950:
5947:
5937:
5933:
5929:
5908:
5902:
5897:
5893:
5870:
5847:
5841:
5838:
5816:
5806:
5801:
5799:
5795:
5791:
5787:
5766:
5763:
5735:
5732:
5707:
5685:
5677:
5676:dot notation,
5673:
5668:
5652:
5628:
5601:
5594:
5566:
5559:
5525:
5516:
5497:
5494:
5470:
5467:
5444:
5419:
5416:
5391:
5367:
5340:
5333:
5330:
5319:
5315:
5294:
5288:
5278:
5273:
5260:
5254:
5251:
5246:
5243:
5237:
5231:
5228:
5223:
5220:
5214:
5208:
5205:
5200:
5197:
5168:
5162:
5156:
5153:
5150:
5127:
5121:
5118:
5115:
5107:
5103:
5086:
5073:
5067:
5061:
5058:
5054:
5039:
5036:
5032:
5027:
5019:
5015:
5011:
5006:
5001:
4997:
4970:
4964:
4961:
4958:
4938:
4913:
4909:
4905:
4900:
4895:
4891:
4867:
4861:
4855:
4849:
4846:
4842:
4837:
4831:
4828:
4823:
4820:
4809:
4791:
4769:
4743:
4740:
4735:
4732:
4718:
4699:
4693:
4690:
4687:
4665:
4662:
4640:
4637:
4629:
4628:differentials
4625:
4621:
4615:
4605:
4603:
4599:
4598:Stefan Banach
4595:
4591:
4587:
4583:
4579:
4574:
4560:
4557:
4554:
4532:
4528:
4524:
4520:
4516:
4510:
4504:
4496:
4480:
4477:
4474:
4452:
4449:
4427:
4407:
4387:
4367:
4347:
4325:
4322:
4319:
4292:
4284:
4278:
4272:
4264:
4247:
4244:
4241:
4221:
4201:
4198:
4195:
4175:
4155:
4135:
4132:
4129:
4109:
4089:
4086:
4083:
4063:
4043:
4023:
4003:
3983:
3961:
3939:
3931:
3930:step function
3915:
3895:
3875:
3867:
3864:must also be
3851:
3829:
3819:
3803:
3786:
3778:
3769:
3762:
3748:
3731:
3728:
3725:
3722:
3720:
3711:
3707:
3704:
3701:
3698:
3695:
3691:
3687:
3684:
3681:
3679:
3670:
3663:
3660:
3653:
3645:
3642:
3633:
3627:
3624:
3619:
3616:
3613:
3610:
3607:
3600:
3596:
3593:
3590:
3588:
3579:
3573:
3570:
3563:
3555:
3552:
3546:
3543:
3540:
3537:
3534:
3531:
3525:
3521:
3518:
3515:
3513:
3504:
3498:
3495:
3488:
3484:
3480:
3475:
3467:
3464:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3435:
3431:
3424:
3420:
3417:
3414:
3412:
3404:
3397:
3394:
3367:
3363:
3359:
3353:
3347:
3339:
3323:
3301:
3298:
3275:
3269:
3266:
3258:
3252:
3249:
3243:
3240:
3237:
3234:
3228:
3222:
3218:
3215:
3212:
3206:
3199:
3196:
3172:
3166:
3146:
3138:
3134:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3092:
3088:
3084:
3080:
3064:
3056:
3055:infinitesimal
3037:
3028:
3025:
3020:
3017:
3000:
2986:
2966:
2958:
2942:
2916:
2913:
2910:
2904:
2901:
2898:
2895:
2892:
2863:
2857:
2854:
2851:
2826:
2797:
2792:
2788:
2784:
2780:
2777:
2772:
2768:
2764:
2761:
2757:
2752:
2748:
2744:
2740:
2737:
2726:
2717:
2713:
2709:
2702:
2693:
2677:
2674:
2671:
2665:
2658:
2655:
2632:
2629:
2622:the limit is
2609:
2589:
2586:
2564:
2542:
2535:. The closer
2522:
2519:
2516:
2496:
2493:
2490:
2487:
2484:
2481:
2476:
2470:
2466:
2462:
2457:
2453:
2449:
2446:
2443:
2440:
2437:
2432:
2428:
2421:
2416:
2410:
2406:
2402:
2397:
2389:
2386:
2383:
2374:
2369:
2362:
2356:
2353:
2347:
2344:
2341:
2335:
2310:
2306:
2302:
2296:
2290:
2270:
2261:
2247:
2225:
2200:
2193:
2190:
2166:
2159:
2156:
2135:
2115:
2093:
2084:
2083:derivative of
2080:
2063:
2060:
2039:
2019:
1999:
1991:
1975:
1967:
1966:
1947:
1925:
1922:
1915:by (or over)
1900:
1897:
1873:
1851:
1831:
1805:
1796:
1793:
1788:
1785:
1757:
1733:
1706:
1699:
1696:
1673:
1653:
1633:
1613:
1605:
1589:
1567:
1545:
1536:
1534:
1530:
1514:
1511:
1508:
1504:
1498:
1491:
1485:
1482:
1476:
1473:
1470:
1464:
1458:
1455:
1451:
1427:
1424:
1421:
1415:
1395:
1392:
1389:
1369:
1366:
1358:
1333:
1313:
1291:
1282:
1264:
1257:
1251:
1248:
1242:
1239:
1236:
1230:
1222:
1216:
1208:
1205:
1198:
1180:
1170:
1169:open interval
1166:
1150:
1142:
1123:
1117:
1110:
1095:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1064:
1062:
1058:
1054:
1050:
1046:
1045:differentials
1042:
1038:
1033:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
987:
982:
980:
975:
973:
968:
967:
965:
964:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
923:
915:
914:
907:
904:
902:
899:
897:
894:
892:
889:
888:
880:
879:
868:
865:
863:
860:
858:
855:
854:
853:
852:
842:
841:
830:
827:
825:
822:
820:
817:
815:
812:
810:
809:Line integral
807:
805:
802:
800:
797:
796:
795:
794:
790:
789:
784:
781:
779:
776:
774:
771:
769:
766:
765:
764:
763:
759:
758:
752:
751:Multivariable
746:
745:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
705:
704:
703:
699:
698:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
664:
663:
662:
656:
650:
649:
638:
635:
633:
630:
628:
625:
623:
620:
618:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
588:
587:
586:
583:
580:
579:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
548:
544:
541:
540:
539:
538:
532:
526:
525:
514:
511:
509:
506:
504:
501:
499:
496:
493:
489:
486:
484:
481:
478:
474:
470:
469:trigonometric
466:
463:
461:
458:
456:
453:
451:
448:
447:
446:
445:
441:
440:
435:
432:
430:
427:
425:
422:
420:
417:
414:
410:
407:
405:
402:
401:
400:
399:
395:
394:
389:
386:
384:
381:
379:
376:
375:
374:
373:
367:
361:
360:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
300:
299:
298:
295:
292:
291:
286:
283:
281:
280:Related rates
278:
276:
273:
271:
268:
266:
263:
261:
258:
257:
256:
255:
251:
250:
243:
240:
238:
237:of a function
235:
233:
232:infinitesimal
230:
229:
228:
225:
222:
218:
215:
214:
213:
212:
208:
207:
201:
195:
194:
188:
185:
183:
180:
178:
175:
174:
169:
166:
164:
161:
160:
156:
153:
152:
151:
150:
131:
125:
122:
116:
110:
107:
104:
101:
94:
87:
84:
78:
73:
69:
60:
59:
56:
53:
52:
48:
47:
44:
40:
33:
19:
17916:
17748:
17737:
17579:Secant cubed
17504:
17497:
17478:Isaac Newton
17448:Brook Taylor
17115:Derivatives
17086:Shell method
16814:Differential
16798:
16626:
16606:
16603:"Derivative"
16589:Khan Academy
16575:
16572:"Derivative"
16549:, Springer,
16545:
16519:
16499:
16486:September 9,
16484:. Retrieved
16462:
16439:, OpenStax,
16435:
16411:
16388:
16369:
16360:
16337:
16317:
16293:, Springer,
16289:
16262:
16244:
16225:, New York:
16222:
16203:
16183:
16161:
16160:(in Czech),
16157:
16134:(2): 69–76,
16131:
16130:(in Czech),
16127:
16094:
16073:
16050:, Springer,
16046:
16026:
16006:
15992:
15968:, Springer,
15964:
15937:
15907:
15888:
15870:
15840:
15808:, Springer,
15805:
15777:
15773:
15743:
15710:
15684:
15675:
15656:
15621:
15617:
15596:
15569:
15563:
15540:
15537:Studia Math.
15536:
15494:
15473:
15450:Barbeau 1961
15445:
15429:
15414:Kolchin 1973
15409:
15393:
15378:Azegami 2020
15373:
15357:
15345:
15338:Roussos 2014
15333:
15317:
15305:
15268:
15256:
15244:
15228:
15204:
15193:Stewart 2002
15188:
15173:Stewart 2002
15168:
15161:Stewart 2002
15146:Stewart 2002
15141:
15125:
15109:
15104:, p. 5.
15097:
15086:Apostol 1967
15081:
15066:Apostol 1967
15060:
14976:
14969:Apostol 1967
14950:Apostol 1967
14945:
14933:
14928:, p. 1.
14917:
14910:Apostol 1967
14897:
14881:
14874:Goodman 1963
14865:
14849:
14623:Stewart 2002
14590:
14561:
14549:
14542:Apostol 1967
14537:
14521:
14509:
14502:Rychlík 1923
14489:
14465:
14441:
14417:
14406:Keisler 2012
14401:
14394:Keisler 2012
14389:
14377:
14370:Stewart 2002
14361:
14349:
14329:
14309:
14293:
14278:Stewart 2002
14274:Apostol 1967
14269:
14254:Stewart 2002
14250:Apostol 1967
14245:
14169:, and so on.
14125:Banach space
14121:vector space
13967:, is then a
13792:
13520:
13068:
12354:starting at
12159:
11949:
11450:
11435:vector field
10859:
10260:
10256:
10252:
10226:
9985:
9478:vector space
9450:
9422:acceleration
9407:
9392:
9388:
9308:
9200:
8951:
8950:
8758:product rule
8292:
8129:
8119:
7986:
7756:Product rule
7754:
7587:
7494:
7449:
7133:
6850:
6473:
6398:
6346:
6336:
6332:
6329:
5927:
5804:
5802:
5675:
5669:
5313:
5274:
4617:
4578:almost every
4575:
3795:
3784:
3336:denotes the
3091:real numbers
3006:
2816:
2712:tangent line
2262:
2082:
2078:
1963:
1603:
1537:
1106:
1065:
1061:acceleration
1034:
1029:
1025:
1014:tangent line
1001:
995:
465:Substitution
227:Differential
216:
200:Differential
43:
17744:Integration
17647:of surfaces
17398:and numbers
17360:Dirichlet's
17330:Telescoping
17283:Alternating
16871:L'Hôpital's
16668:Precalculus
16471:Pearson PLC
15740:John, Fritz
15624:(1): 1–46,
15402:84–85
15398:Funaro 1992
15322:Davvaz 2023
15294:Davvaz 2023
15273:Guzman 2003
15102:Warner 1983
14938:Cajori 1923
14890:Cajori 2007
14554:Cajori 2007
14528:, cited in
14526:Banach 1931
14498:Jarník 1922
14480:, pp.
14470:Gonick 2012
14446:Gonick 2012
14422:Gonick 2012
14354:Gonick 2012
14334:Gonick 2012
14314:Gonick 2012
14298:Gonick 2012
13022:pushforward
12832:. However,
11762:, then the
10255:called the
6476:exponential
5515:superscript
5435:, read as "
5358:, read as "
4596:. In 1931,
4380:is one; if
3133:reciprocals
1724:, read as "
1281:real number
1171:containing
1143:at a point
998:mathematics
926:Precalculus
919:Miscellanea
884:Specialized
791:Definitions
558:Alternating
396:Definitions
209:Definitions
17932:Categories
17769:stochastic
17443:Adequality
17129:Divergence
17002:Arc length
16799:Derivative
16458:Hass, Joel
15934:Gbur, Greg
15787:1711.10349
15588:0101.03702
15460:References
15436:, p.
15416:, p.
15400:, p.
15364:, p.
15324:, p.
15296:, p.
15275:, p.
15235:, p.
15211:, p.
15195:, p.
15175:, p.
15132:, p.
15116:, p.
14922:Evans 1999
14904:, p.
14856:, p.
14514:David 2018
14494:Jašek 1922
14456:, p.
14432:, p.
14340:, p.
14320:, p.
14284:, p.
14260:, p.
14211:Derivation
14140:continuous
14123:, such as
13969:linear map
12610:such that
10919:be such a
9436:See also:
9399:polynomial
8754:chain rule
8131:Chain rule
5805:D-notation
5786:arc length
5277:prime mark
5106:chain rule
4630:, such as
4602:meager set
3952:less than
3866:continuous
1968:below. If
1604:derivative
1346:such that
1195:, and the
1103:As a limit
1098:Definition
1049:prime mark
1002:derivative
906:Variations
901:Stochastic
891:Fractional
760:Formalisms
723:Divergence
692:Identities
672:Divergence
217:Derivative
168:Continuity
17881:Functions
17642:of curves
17637:Curvature
17524:Integrals
17318:Maclaurin
17298:Geometric
17189:Geometric
17139:Laplacian
16851:linearity
16691:Factorial
16608:MathWorld
16582:EMS Press
16343:CRC Press
15832:259885793
15729:872345701
15522:226442409
15380:. See p.
15350:Gbur 2011
15249:Gbur 2011
13908:→
13467:∂
13452:∂
13319:…
13115:exist at
12727:‖
12719:‖
12714:‖
12664:−
12639:‖
12628:→
12586:→
12571::
12410:≈
12119:∂
12111:∂
12078:∑
11913:−
11877:→
11732:…
11405:…
11383:∇
11347:…
11308:∇
11197:…
11165:∂
11157:∂
11148:…
11126:…
11094:∂
11086:∂
11056:…
11034:∇
10996:…
10891:…
10821:…
10802:…
10780:−
10761:…
10736:…
10706:→
10676:…
10644:∂
10636:∂
10597:…
10518:…
10455:∂
10447:∂
10416:∂
10408:∂
10206:∂
10198:∂
10166:∂
10162:∂
10131:∂
10028:…
9864:−
9833:→
9657:…
9560:…
9330:. If the
9179:−
8889:
8857:
8825:
8714:
8708:−
8685:−
8664:
8576:
8570:−
8536:
8519:−
8498:
8446:−
8359:
8353:−
8332:
8251:⋅
8041:−
7960:⋅
7944:α
7923:α
7915:whenever
7895:α
7878:α
7735:β
7713:α
7645:β
7631:α
7614:β
7605:α
7391:
7337:−
7304:−
7292:−
7277:
7223:−
7190:−
7166:
7108:
7074:
7043:
7018:
6970:
6964:−
6949:
6901:
6883:
6790:
6763:
6669:
6595:
6484:logarithm
6452:−
6270:∂
6266:∂
6248:∂
6244:∂
6152:∂
6148:∂
6130:∂
6126:∂
5767:¨
5736:˙
5382:prime of
5316:, due to
5238:⋅
4450:−
3688:
3614:⋅
3597:
3538:⋅
3522:
3481:−
3450:⋅
3421:
3250:−
3219:
3113:⋯
3087:extension
2781:
2741:
2520:≠
2463:−
2403:−
2354:−
2182:whenever
1748:prime of
1512:ε
1483:−
1459:−
1393:≠
1370:δ
1314:δ
1292:ε
1249:−
1220:→
896:Malliavin
783:Geometric
682:Laplacian
632:Dirichlet
543:Geometric
123:−
70:∫
17906:Infinity
17759:ordinary
17739:Calculus
17632:Manifold
17365:Integral
17308:Infinite
17303:Harmonic
17288:Binomial
17134:Gradient
17077:Volumes
16888:Quotient
16829:Notation
16660:Calculus
16543:(1983),
16520:Calculus
16460:(2014).
16412:Calculus
16149:(1922),
16004:(2012),
15936:(2011),
15869:(1999),
15531:(1931),
15310:Lee 2013
14761:′
14226:Integral
14200:See also
14189:integers
14131:and the
13411:′
13179:′
12991:′
12844:′
12691:′
12556:′
12478:′
12434:′
11235:gradient
10109:′
9959:′
9809:′
9418:velocity
9124:‴
9093:″
9066:″
9035:′
9008:′
8412:′
8258:′
8229:′
8209:′
8051:′
8034:′
8020:′
7947:′
7902:′
7888:′
7810:′
7793:′
7782:′
7652:′
7638:′
7624:′
7589:Sum rule
7555:′
5619:for the
5498:‴
5471:″
5420:′
5334:′
4931:for the
4608:Notation
4586:monotone
3908:and let
3398:′
3316:, where
3200:′
3188:becomes
3083:infinite
2659:′
2194:′
2160:′
2064:′
1700:′
1057:velocity
1006:function
936:Glossary
846:Advanced
824:Jacobian
778:Exterior
708:Gradient
700:Theorems
667:Gradient
606:Integral
568:Binomial
553:Harmonic
413:improper
409:Integral
366:Integral
348:Reynolds
323:Quotient
252:Concepts
88:′
55:Calculus
17764:partial
17569:inverse
17557:inverse
17483:Fluxion
17293:Fourier
17159:Stokes'
17154:Green's
16876:Product
16736:Tangent
16584:, 2001
15942:Bibcode
15648:1967725
15053:
15033:
15029:
15009:
15005:
14985:
14838:
14818:
14784:
14733:
14482:237–238
14191:by the
14110:
14090:
14062:
14033:
13965:
13945:
13889:
13860:
13687:
13658:
13583:
13561:
13347:
13276:
13252:
13230:
13165:
13143:
13139:
13117:
13093:
13071:
12979:, then
12977:
12957:
12904:
12875:
12801:
12772:
12378:
12356:
12310:
12284:
12240:
12211:
11994:
11972:
11760:
11700:
11696:
11670:
11608:in the
11586:
11566:
11502:
11473:
11024:
10972:
10335:
10265:
9945:, then
9943:
9923:
9414:physics
9385:
9358:
9285:
9252:
9223:
9203:
9197:
9165:
9161:
9141:
9137:
9112:
9106:is the
9079:
9054:
9048:is the
9021:
8996:
8195:, then
8193:
8140:
7865:
7845:
7747:
7727:
7539:
7519:
6391:2.71828
6314:
6197:
6080:
6035:
5981:
5940:
5924:
5885:
5829:
5809:
5794:physics
5782:
5753:
5720:
5700:
5674:or the
5665:
5645:
5641:
5621:
5584:
5551:
5511:
5486:
5457:
5437:
5433:
5408:
5404:
5384:
5380:
5360:
5356:
5322:
5310:
5281:
4804:
4784:
4760:
4721:
4678:
4655:
4465:
4442:
4338:
4312:
3974:
3954:
3928:be the
3844:, then
3842:
3822:
3314:
3291:
3089:of the
2957:tangent
2839:
2819:
2690:
2647:
2577:
2557:
2238:
2218:
2148:equals
2106:
2086:
2081:or the
1962:". See
1960:
1940:
1913:
1890:
1886:
1866:
1822:
1774:
1770:
1750:
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6375:e
6355:a
6299:)
6294:)
6291:y
6288:,
6285:x
6282:(
6279:f
6273:x
6259:(
6251:x
6239:=
6236:)
6233:y
6230:,
6227:x
6224:(
6221:f
6216:2
6211:x
6207:D
6181:)
6176:)
6173:y
6170:,
6167:x
6164:(
6161:f
6155:x
6141:(
6133:y
6121:=
6118:)
6115:y
6112:,
6109:x
6106:(
6103:f
6098:y
6095:x
6091:D
6068:)
6065:y
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6059:x
6056:(
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6048:x
6044:D
6021:u
6016:x
6012:D
5991:x
5969:)
5966:y
5963:,
5960:x
5957:(
5954:f
5951:=
5948:u
5912:)
5909:x
5906:(
5903:f
5898:n
5894:D
5871:n
5851:)
5848:x
5845:(
5842:f
5839:D
5817:D
5764:y
5733:y
5708:t
5686:y
5653:f
5629:n
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5602:n
5599:(
5595:f
5570:)
5567:4
5564:(
5560:f
5534:v
5531:i
5526:f
5495:f
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5417:y
5392:x
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5341:x
5338:(
5331:f
5298:)
5295:x
5292:(
5289:f
5261:.
5255:x
5252:d
5247:u
5244:d
5232:u
5229:d
5224:y
5221:d
5215:=
5209:x
5206:d
5201:y
5198:d
5175:)
5172:)
5169:x
5166:(
5163:g
5160:(
5157:f
5154:=
5151:y
5131:)
5128:x
5125:(
5122:g
5119:=
5116:u
5087:.
5082:)
5077:)
5074:x
5071:(
5068:f
5062:x
5059:d
5055:d
5048:(
5040:x
5037:d
5033:d
5028:=
5020:2
5016:x
5012:d
5007:y
5002:2
4998:d
4974:)
4971:x
4968:(
4965:f
4962:=
4959:y
4939:n
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4910:x
4906:d
4901:y
4896:n
4892:d
4868:.
4865:)
4862:x
4859:(
4856:f
4850:x
4847:d
4843:d
4838:=
4832:x
4829:d
4824:y
4821:d
4792:x
4770:y
4744:x
4741:d
4736:y
4733:d
4703:)
4700:x
4697:(
4694:f
4691:=
4688:y
4666:x
4663:d
4641:y
4638:d
4561:0
4558:=
4555:x
4533:3
4529:/
4525:1
4521:x
4517:=
4514:)
4511:x
4508:(
4505:f
4481:0
4478:=
4475:x
4453:1
4428:h
4408:0
4388:h
4368:h
4348:h
4326:0
4323:=
4320:x
4297:|
4293:x
4289:|
4285:=
4282:)
4279:x
4276:(
4273:f
4248:h
4245:+
4242:a
4222:a
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4199:+
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4110:a
4090:h
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4024:f
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3962:a
3940:x
3916:f
3896:a
3876:a
3852:f
3830:a
3804:f
3785:x
3732:.
3729:x
3726:2
3723:=
3712:)
3708:x
3705:d
3702:+
3699:x
3696:2
3692:(
3682:=
3671:)
3664:x
3661:d
3654:2
3650:)
3646:x
3643:d
3640:(
3634:+
3628:x
3625:d
3620:x
3617:d
3611:x
3608:2
3601:(
3591:=
3580:)
3574:x
3571:d
3564:2
3560:)
3556:x
3553:d
3550:(
3547:+
3544:x
3541:d
3535:x
3532:2
3526:(
3516:=
3505:)
3499:x
3496:d
3489:2
3485:x
3476:2
3472:)
3468:x
3465:d
3462:(
3459:+
3456:x
3453:d
3447:x
3444:2
3441:+
3436:2
3432:x
3425:(
3415:=
3408:)
3405:x
3402:(
3395:f
3368:2
3364:x
3360:=
3357:)
3354:x
3351:(
3348:f
3302:x
3299:d
3276:)
3270:x
3267:d
3262:)
3259:x
3256:(
3253:f
3247:)
3244:x
3241:d
3238:+
3235:x
3232:(
3229:f
3223:(
3213:=
3210:)
3207:x
3204:(
3197:f
3176:)
3173:x
3170:(
3167:f
3147:d
3119:1
3116:+
3110:+
3107:1
3104:+
3101:1
3065:f
3041:)
3038:a
3035:(
3029:x
3026:d
3021:f
3018:d
2987:a
2967:f
2943:h
2923:)
2920:)
2917:h
2914:+
2911:a
2908:(
2905:f
2902:,
2899:h
2896:+
2893:a
2890:(
2870:)
2867:)
2864:a
2861:(
2858:f
2855:,
2852:a
2849:(
2827:f
2798:)
2793:2
2789:x
2785:(
2773:2
2769:x
2765:2
2762:+
2758:)
2753:2
2749:x
2745:(
2678:x
2675:2
2672:=
2669:)
2666:x
2663:(
2656:f
2633:a
2630:2
2610:a
2590:a
2587:2
2565:0
2543:h
2523:0
2517:h
2497:.
2494:h
2491:+
2488:a
2485:2
2482:=
2477:h
2471:2
2467:a
2458:2
2454:h
2450:+
2447:h
2444:a
2441:2
2438:+
2433:2
2429:a
2422:=
2417:h
2411:2
2407:a
2398:2
2394:)
2390:h
2387:+
2384:a
2381:(
2375:=
2370:h
2366:)
2363:a
2360:(
2357:f
2351:)
2348:h
2345:+
2342:a
2339:(
2336:f
2311:2
2307:x
2303:=
2300:)
2297:x
2294:(
2291:f
2271:f
2248:f
2226:f
2204:)
2201:a
2198:(
2191:f
2170:)
2167:a
2164:(
2157:f
2136:a
2116:f
2094:f
2061:f
2040:x
2020:f
2000:x
1976:f
1948:a
1926:x
1923:d
1901:f
1898:d
1874:a
1852:x
1832:f
1809:)
1806:a
1803:(
1797:x
1794:d
1789:f
1786:d
1758:a
1734:f
1710:)
1707:a
1704:(
1697:f
1674:a
1654:f
1634:a
1614:f
1590:L
1568:a
1546:f
1515:,
1505:|
1499:h
1495:)
1492:a
1489:(
1486:f
1480:)
1477:h
1474:+
1471:a
1468:(
1465:f
1456:L
1452:|
1431:)
1428:h
1425:+
1422:a
1419:(
1416:f
1396:0
1390:h
1363:|
1359:h
1355:|
1334:h
1265:h
1261:)
1258:a
1255:(
1252:f
1246:)
1243:h
1240:+
1237:a
1234:(
1231:f
1223:0
1217:h
1209:=
1206:L
1181:a
1151:a
1127:)
1124:x
1121:(
1118:f
985:e
978:t
971:v
549:)
494:)
490:(
479:)
415:)
223:)
135:)
132:a
129:(
126:f
120:)
117:b
114:(
111:f
108:=
105:t
102:d
98:)
95:t
92:(
85:f
79:b
74:a
41:.
34:.
20:)
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