1863:
5911:
3346:
1969:
1508:
2925:
20:
3341:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {x^{2}+2x\Delta x+(\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}2x+\Delta x\\\end{aligned}}}
4921:
5319:. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.
1143:
2862:
provided such a limit exists. We have thus succeeded in properly defining the derivative of a function, meaning that the 'slope of the tangent line' now has a precise mathematical meaning. Differentiating a function using the above definition is known as differentiation from first principles. Here
6182:
Newton began his work in 1665 and
Leibniz began his in 1676. However, Leibniz published his first paper in 1684, predating Newton's publication in 1693. It is possible that Leibniz saw drafts of Newton's work in 1673 or 1676, or that Newton made use of Leibniz's work to refine his own. Both Newton
1759:
vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point. The slope of a curve at a particular point is equal to the
3980:, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known.
5337:
The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function
4365:
of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a
6122:
The term infinitesimal can sometimes lead people to wrongly believe there is an 'infinitely small number'—i.e. a positive real number that is smaller than any other real number. In fact, the term 'infinitesimal' is merely a shorthand for a limiting process. For this reason,
4384:
One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the
4337:
This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.
5306:
has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of
2857:
4868:
4024:" Isaac Barrow is generally given credit for the early development of the derivative. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to
3995:
relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes. For their ideas on derivatives, both Newton and
Leibniz built on significant earlier work by mathematicians such as
1623:
1441:
2453:
2069:. Because these two points are fairly close together, the dotted line and tangent line have a similar slope. As the two points become closer together, the error produced by the secant line becomes vanishingly small.
2318:
1960:
is then simply the slope of this tangent line. Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as a
1667:
4393:, and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a
2240:
3711:
4670:
4415:" — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in
4592:
126:
5295:
5115:
2930:
3571:
1212:
4778:
4031:
Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as
1704:
2172:
4505:
3917:, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He obtained, for example, that the maximum (for positive
1965:. If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar:
23:
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.
3493:
4968:
6158:
2614:
1726:
being the Greek letter delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs such as
3424:
6091:
6049:
5987:
5032:
2542:
2733:
4793:
3372:
2479:
2344:
2195:
2113:
1547:
1479:
1317:
2894:
2067:
1897:
1791:
1757:
1724:
4994:
2696:
2577:
1352:
1179:
2034:
2002:
1929:
4334:
must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.
2725:
2508:
1246:
5472:, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients
1837:
1272:
1205:
3991:(1646–1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the
3447:
2917:
2661:
1502:
6111:
6007:
3611:
3591:
3392:
2634:
2364:
1949:
1857:
1811:
1392:
1372:
2481:
gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'. The value that is being approached is the derivative of
1555:
6368:
6169:
Not every function can be differentiated, hence why the definition only applies if 'the limit exists'. For more information, see the
Knowledge article on
5947:
is somewhat involved, it is easy to appreciate what a function is intuitively. A function takes an input and produces an output. For example, the function
4022:
I had the hint of this method from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general.
1397:
5934:
This is not a formal definition of what a tangent line is. The definition of the derivative as a limit makes rigorous this notion of tangent line.
5701:. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called
2372:
4432:
is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.
6197:
4058:, besides extending integral calculus to many more functions, clarified the relation between derivation and integration with the notion of
2248:
4701:. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example,
4322:
Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in
1631:
222:
4194:
is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.
1319:. In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form
5828:
looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of
2200:
6347:
3902:(1114–1185); indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as "
3624:
1078:
6287:
4405:
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called
4370:", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.
5989:
takes a number and squares it. The number that the function performs an operation on is often represented using the letter
5300:
In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that
984:
that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being
483:
463:
3717:, meaning that sometimes further techniques are needed to find the derivative of a function. These techniques include the
6184:
4693:
is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A
2197:
is a small number. As before, the slope of the line passing through these two points can be calculated with the formula
2074:
The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph
1862:
6543:
6457:
4705:, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation
3498:
959:
522:
4787:
in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation
4516:
45:
5216:
5121:
The mean value theorem gives a relationship between values of the derivative and values of the original function. If
5037:
4711:
3894:
478:
201:
4603:
2366:, the slope of the secant line gets closer and closer to the slope of the tangent line. This is formally written as
468:
4409:. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "
1672:
1354:. The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in
3992:
1039:
799:
473:
453:
135:
6365:
2118:
6563:
5893:
4690:
4442:
4083:
3812:
changes in several directions at once. Taking the best linear approximation in a single direction determines a
5855:, one of these two functions has a graph that looks like the circle. (These two functions also happen to meet
5705:. It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there exist
4694:
4054:
The 20th century brought two major steps towards our present understanding and practice of derivation :
581:
528:
414:
3910:
5903:
4162:
240:
212:
4426:
is the derivative (with respect to time) of an object's displacement (distance from the original position)
3892:, although these were primarily used to study areas and volumes rather than derivatives and tangents (see
323:
5819:
5500:
3452:
832:
445:
283:
255:
4927:
5883:
4323:
3737:
1000:
703:
667:
449:
328:
217:
207:
6126:
2582:
5888:
5718:
3988:
1043:
6342:
308:
6352:
6230:
6187:
over who first invented calculus, which shook the mathematical community in the early 18th century.
5867:
3889:
602:
167:
6196:
This was a monumental achievement, even though a restricted version had been proven previously by
4689:
A differential equation is a relation between a collection of functions and their derivatives. An
2852:{\displaystyle {\frac {dy}{dx}}=f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}
6529:
6183:
and
Leibniz claimed that the other plagiarized their respective works. This resulted in a bitter
5898:
4863:{\displaystyle {\frac {\partial u}{\partial t}}=\alpha {\frac {\partial ^{2}u}{\partial x^{2}}}.}
4106:
4063:
3397:
1027:
of a single real variable, the derivative of a function at a point generally determines the best
1004:
916:
708:
597:
6054:
6012:
5950:
5154:, then the mean value theorem says that under mild hypotheses, the slope between the two points
4999:
1073:
equals the force applied to the body; rearranging this derivative statement leads to the famous
6254:
5944:
4702:
4379:
4306:
3614:
1957:
1101:
996:
952:
881:
842:
726:
662:
586:
3354:
2461:
2326:
2177:
2077:
1514:
1446:
1281:
6332:
5564:, and higher-degree coefficients, these coefficients are determined by higher derivatives of
5383:, and it may be possible to get a better approximation by considering a quadratic polynomial
4684:
4406:
4386:
4327:
4310:
4032:
2866:
2039:
1869:
1763:
1729:
1709:
1117:
1093:, derivatives determine the most efficient ways to transport materials and design factories.
1054:
1003:, and their applications. The derivative of a function at a chosen input value describes the
926:
592:
368:
313:
274:
4973:
2666:
2547:
1322:
1149:
6328:
4346:
4071:
4055:
2513:
2007:
1975:
1902:
1028:
1024:
931:
911:
837:
506:
430:
404:
318:
2701:
2484:
1222:
8:
6338:
5871:
5728:
5667:
4059:
4025:
3881:
3859:
1816:
1251:
1207:, meaning at that exact point, the slope of the curve and the straight line are the same.
1184:
1113:
1090:
1020:
906:
876:
866:
753:
607:
409:
265:
148:
143:
6486:
Berggren, J. L. (1990). "Innovation and
Tradition in Sharaf al-Din al-Tusi's Muadalat".
5834:. The circle, for instance, can be pasted together from the graphs of the two functions
4043:(1815–1897). It was also during this period that the differentiation was generalized to
3898:). The use of infinitesimals to compute rates of change was developed significantly by
3429:
2899:
2643:
1618:{\displaystyle {\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {-6}{+3}}=-2}
1484:
6523:
6503:
6413:
6405:
6096:
5992:
5915:
4915:
4698:
4009:
3903:
3813:
3596:
3576:
3377:
2619:
2349:
1934:
1842:
1796:
1377:
1357:
1105:
1023:
at that point, provided that the derivative exists and is defined at that point. For a
871:
774:
758:
698:
693:
688:
652:
533:
457:
363:
358:
162:
157:
1108:. Derivatives and their generalizations appear in many fields of mathematics, such as
6568:
6453:
6417:
6283:
6170:
5910:
5702:
5328:
4697:
is a differential equation that relates functions of more than one variable to their
4416:
4354:
4216:
4075:
3730:
1436:{\displaystyle {\text{slope }}={\frac {{\text{ change in }}y}{{\text{change in }}x}}}
1097:
1086:
1035:
1007:
of the function near that input value. The process of finding a derivative is called
985:
945:
779:
557:
440:
393:
250:
245:
4005:
6495:
6397:
6201:
4342:
4168:
4067:
4040:
4036:
3997:
3847:
1125:
1109:
789:
683:
657:
518:
435:
399:
1968:
6547:
6372:
5706:
4411:
4394:
4331:
4044:
3729:. Other functions cannot be differentiated at all, giving rise to the concept of
921:
794:
748:
743:
630:
543:
488:
6304:
4028:, while Leibniz systematically developed much of the notation still used today.
2863:
is a proof, using differentiation from first principles, that the derivative of
5666:, and its coefficients can be found by a generalization of the above formulas.
4362:
4078:) and became fundamental to nowadays applied analysis especially by the use of
4020:(1616–1703). Regarding Fermat's influence, Newton once wrote in a letter that "
3869:
1121:
804:
612:
384:
3899:
6557:
5332:
4784:
4128:
4124:
4114:
4079:
4048:
4013:
3977:
3726:
2637:
1211:
1082:
784:
548:
303:
260:
6541:
1507:
1049:
Differentiation has applications in nearly all quantitative disciplines. In
4429:
4367:
4001:
3984:
3865:
3722:
2448:{\displaystyle \lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}
1062:
538:
288:
1142:
4017:
1962:
973:
901:
1061:
of the body, and the derivative of the velocity with respect to time is
6409:
5794:
5359:
4358:
3885:
3877:
3718:
3714:
3618:
1137:
992:
647:
571:
298:
293:
197:
6507:
4906:
is a constant that depends on how fast heat diffuses through the rod.
3713:. However, many other functions cannot be differentiated as easily as
5697:
The limit of the Taylor polynomials is an infinite series called the
4357:
test can still be used to analyse critical points by considering the
4259:
could be a local minimum, a local maximum, or neither. (For example,
2313:{\displaystyle {\text{slope}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}
576:
566:
19:
6401:
1662:{\displaystyle {\frac {{\text{change in }}y}{{\text{change in }}x}}}
6499:
4423:
4390:
4350:
1066:
1058:
981:
642:
389:
346:
35:
6384:
Broadbent, T. A. A.; Kline, M. (October 1968). "Reviewed work(s):
3736:
A closely related concept to the derivative of a function is its
1050:
1016:
5863:, but this is not guaranteed by the implicit function theorem.)
3800:
are vectors, then the best linear approximation to the graph of
1839:
because the slope of the tangent line to that point is equal to
6200:(1638–1675), and some key examples can be found in the work of
5724:
3873:
1215:
The derivative at different points of a differentiable function
991:
The primary objects of study in differential calculus are the
1275:
1042:. This states that differentiation is the reverse process to
1012:
4397:
and it, too, can be found using the calculus of variations.
4280:, but it has neither a maximum nor a minimum there, whereas
2235:{\displaystyle {\text{slope }}={\frac {\Delta y}{\Delta x}}}
5797:. The implicit function theorem converts relations such as
5670:
gives a precise bound on how good the approximation is. If
4436:
For example, if an object's position on a line is given by
4203:
is twice differentiable, then conversely, a critical point
1070:
4070:) extended derivation to generalized functions (e.g., the
3983:
The modern development of calculus is usually credited to
1100:
of a function. Equations involving derivatives are called
4924:
The mean value theorem: For each differentiable function
3706:{\displaystyle {\frac {d}{dx}}(5x^{4})=5(4)x^{3}=20x^{3}}
6160:
is not a fraction—rather, it is the limit of a fraction.
6009:, but there is no difference whatsoever between writing
5866:
The implicit function theorem is closely related to the
5847:. In a neighborhood of every point on the circle except
2663:
represents an infinitesimal change in x. In summary, if
6431:
6429:
6427:
6282:. New York: Oxford University Press. pp. 155–157.
5322:
4920:
5062:
6129:
6099:
6057:
6015:
5995:
5953:
5219:
5040:
5002:
4976:
4930:
4796:
4714:
4606:
4519:
4445:
3627:
3599:
3579:
3501:
3455:
3432:
3400:
3380:
3357:
2928:
2902:
2869:
2736:
2704:
2669:
2646:
2622:
2585:
2550:
2516:
2487:
2464:
2375:
2352:
2329:
2251:
2203:
2180:
2121:
2080:
2042:
2010:
1978:
1937:
1905:
1872:
1845:
1819:
1799:
1766:
1732:
1712:
1675:
1634:
1558:
1517:
1487:
1449:
1400:
1380:
1360:
1325:
1284:
1254:
1225:
1187:
1152:
48:
6424:
5870:, which states when a function looks like graphs of
5313:
must equal the slope of one of the tangent lines of
6152:
6105:
6085:
6043:
6001:
5981:
5496:that makes the approximation as good as possible.
5289:
5109:
5026:
4988:
4962:
4862:
4772:
4664:
4586:
4499:
4301:and a minimum and a maximum, respectively, there.)
3705:
3605:
3585:
3565:
3487:
3441:
3418:
3386:
3366:
3340:
2911:
2888:
2851:
2719:
2690:
2655:
2628:
2608:
2571:
2536:
2502:
2473:
2447:
2358:
2338:
2312:
2234:
2189:
2166:
2107:
2061:
2028:
1996:
1943:
1923:
1891:
1851:
1831:
1805:
1785:
1751:
1718:
1698:
1661:
1617:
1541:
1496:
1473:
1435:
1386:
1366:
1346:
1311:
1266:
1240:
1199:
1173:
1011:. Geometrically, the derivative at a point is the
120:
6336:
4661:
4583:
4496:
4389:is not immediately clear. These paths are called
3764:is the slope of the tangent line to the graph of
3566:{\displaystyle {\frac {d(ax^{n})}{dx}}=anx^{n-1}}
1951:. (The axes of the graph do not use a 1:1 scale.)
1760:slope of the tangent to that point. For example,
6555:
5676:is a polynomial of degree less than or equal to
4587:{\displaystyle {\dot {x}}(t)=x'(t)=-32t+16,\,\!}
3301:
3224:
3121:
3040:
2961:
2781:
2377:
121:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
5290:{\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}
5110:{\displaystyle f'(c)={\tfrac {f(b)-f(a)}{b-a}}}
4773:{\displaystyle F(t)=m{\frac {d^{2}x}{dt^{2}}}.}
6538:Differential Calculus: From Practice to Theory
6383:
4665:{\displaystyle {\ddot {x}}(t)=x''(t)=-32,\,\!}
4089:
3864:The concept of a derivative in the sense of a
5186:is equal to the slope of the tangent line to
3853:
3495:. This proof can be generalised to show that
1057:of a moving body with respect to time is the
953:
5712:
3942:, and concluded therefrom that the equation
1931:. The slope of the tangent line is equal to
1699:{\displaystyle {\frac {\Delta y}{\Delta x}}}
1096:Derivatives are frequently used to find the
6540:. 2022, personal.psu.edu/ecb5/DiffCalc.pdf
6452:. Cambridge University Press. p. 144.
6450:Theories of Light: From Descartes to Newton
5822:, then around most points, the zero set of
5420:. Still better might be a cubic polynomial
6521:
5754:, then the circle is the set of all pairs
4888:is the temperature of the rod at position
2167:{\displaystyle (x+\Delta x,f(x+\Delta x))}
1899:, with a straight line that is tangent to
960:
946:
6386:The History of Ancient Indian Mathematics
6113:is often described as a 'dummy variable'.
4956:
4678:
4660:
4582:
4500:{\displaystyle x(t)=-16t^{2}+16t+32,\,\!}
4495:
4373:
81:
6488:Journal of the American Oriental Society
6485:
6435:
4919:
3846:in all directions at once is called the
2579:, the derivative can also be written as
1972:The dotted line goes through the points
1967:
1861:
1506:
1210:
1141:
988:—the study of the area beneath a curve.
18:
16:Area of mathematics; subarea of calculus
6348:MacTutor History of Mathematics Archive
5723:Some natural geometric shapes, such as
5682:, then the Taylor polynomial of degree
4313:, involves considering the sign of the
3953:has exactly one positive solution when
3868:is a very old one, familiar to ancient
3782:are one-dimensional, the derivative of
484:Differentiating under the integral sign
6556:
6277:
5787:, and is not the same as the graph of
4309:. An alternative approach, called the
3964:, and two positive solutions whenever
3752:are real variables, the derivative of
6447:
6441:
6302:
6231:"Definition of DIFFERENTIAL CALCULUS"
5943:Though the technical definition of a
5781:. This set is called the zero set of
5638:. Using these coefficients gives the
4909:
6528:. London: MacMillan and Co. p.
5323:Taylor polynomials and Taylor series
4319:on each side of the critical point.
5538:the best possible choice is always
5518:the best possible choice is always
4215:can be analysed by considering the
3776:. Because the source and target of
3488:{\displaystyle {\frac {dy}{dx}}=2x}
1146:The graph of an arbitrary function
13:
6536:Boman, Eugene, and Robert Rogers.
5812:into functions. It states that if
5648:. The Taylor polynomial of degree
4963:{\displaystyle f:\to \mathbb {R} }
4841:
4827:
4808:
4800:
3410:
3358:
3328:
3305:
3281:
3263:
3251:
3228:
3204:
3173:
3161:
3125:
3101:
3070:
3044:
3020:
2994:
2965:
2840:
2814:
2785:
2465:
2436:
2410:
2381:
2330:
2301:
2275:
2223:
2215:
2181:
2152:
2131:
1713:
1687:
1679:
1104:and are fundamental in describing
30:Part of a series of articles about
14:
6580:
6255:"Definition of INTEGRAL CALCULUS"
4597:and the object's acceleration is
4349:function is a point at which the
3895:The Method of Mechanical Theorems
1504:, as shown in the diagram below:
6515:
6153:{\displaystyle {\frac {dy}{dx}}}
5909:
2609:{\displaystyle {\frac {dy}{dx}}}
1181:. The orange line is tangent to
6466:
6377:
6358:
6190:
6176:
4094:
3993:fundamental theorem of calculus
2458:The above expression means 'as
1040:fundamental theorem of calculus
1031:to the function at that point.
6479:
6321:
6296:
6271:
6247:
6223:
6163:
6116:
6067:
6061:
6025:
6019:
5963:
5957:
5937:
5928:
5894:Techniques for differentiation
5267:
5261:
5252:
5246:
5234:
5228:
5132:is a real-valued function and
5089:
5083:
5074:
5068:
5055:
5049:
5021:
5009:
4952:
4949:
4937:
4724:
4718:
4691:ordinary differential equation
4645:
4639:
4625:
4619:
4558:
4552:
4538:
4532:
4510:then the object's velocity is
4455:
4449:
4084:partial differential equations
3674:
3668:
3659:
3643:
3524:
3508:
3311:
3270:
3260:
3234:
3180:
3170:
3131:
3077:
3061:
3050:
3015:
3009:
3000:
2985:
2971:
2835:
2829:
2820:
2805:
2791:
2774:
2768:
2714:
2708:
2685:
2679:
2566:
2560:
2531:
2525:
2497:
2491:
2431:
2425:
2416:
2401:
2387:
2296:
2290:
2281:
2266:
2161:
2158:
2143:
2122:
2102:
2099:
2093:
2081:
2036:, which both lie on the curve
2023:
2011:
1991:
1979:
1918:
1906:
1306:
1303:
1297:
1285:
1235:
1229:
1168:
1162:
999:, related notions such as the
115:
109:
100:
94:
78:
72:
1:
6211:
5709:which are also not analytic.
4695:partial differential equation
4330:, a continuous function on a
1131:
1079:Newton's second law of motion
415:Integral of inverse functions
6216:
5904:Notation for differentiation
5654:is the polynomial of degree
3976:. The historian of science,
7:
6280:How to Think about Analysis
5877:
5820:continuously differentiable
4090:Applications of derivatives
3816:, which is usually denoted
3419:{\displaystyle 2x+\Delta x}
833:Calculus on Euclidean space
256:Logarithmic differentiation
10:
6585:
6388:by C. N. Srinivasiengar".
6086:{\displaystyle f(y)=y^{2}}
6044:{\displaystyle f(x)=x^{2}}
5982:{\displaystyle f(x)=x^{2}}
5716:
5326:
5027:{\displaystyle c\in (a,b)}
4913:
4682:
4400:
4377:
3857:
3854:History of differentiation
2346:gets closer and closer to
1135:
1069:of a body with respect to
1034:Differential calculus and
5889:Numerical differentiation
5727:, cannot be drawn as the
5719:Implicit function theorem
5713:Implicit function theorem
4137:, then the derivative of
4074:previously introduced in
3989:Gottfried Wilhelm Leibniz
2698:, then the derivative of
2510:; this can be written as
1077:equation associated with
567:Summand limit (term test)
6390:The Mathematical Gazette
6353:University of St Andrews
6185:controversy between them
5921:
5868:inverse function theorem
5660:which best approximates
4345:, a critical point of a
4294:has a critical point at
4273:has a critical point at
3367:{\displaystyle \Delta x}
2474:{\displaystyle \Delta x}
2339:{\displaystyle \Delta x}
2190:{\displaystyle \Delta x}
2108:{\displaystyle (x,f(x))}
1542:{\displaystyle y=-2x+13}
1474:{\displaystyle y=-2x+13}
1312:{\displaystyle (a,f(a))}
1065:. The derivative of the
1053:, the derivative of the
251:Implicit differentiation
241:Differentiation notation
168:Inverse function theorem
6259:www.merriam-webster.com
6235:www.merriam-webster.com
5899:List of calculus topics
5884:Differential (calculus)
4107:differentiable function
4064:theory of distributions
3872:mathematicians such as
3840:. The linearization of
3617:. This is known as the
2889:{\displaystyle y=x^{2}}
2062:{\displaystyle y=x^{2}}
1892:{\displaystyle y=x^{2}}
1786:{\displaystyle y=x^{2}}
1752:{\displaystyle y=x^{2}}
1719:{\displaystyle \Delta }
709:Helmholtz decomposition
6154:
6107:
6087:
6045:
6003:
5983:
5916:Mathematics portal
5291:
5118:
5111:
5028:
4990:
4989:{\displaystyle a<b}
4964:
4864:
4774:
4679:Differential equations
4666:
4588:
4501:
4407:differential equations
4380:Calculus of variations
4374:Calculus of variations
4307:second derivative test
4149:is zero. Points where
3707:
3607:
3587:
3567:
3489:
3443:
3420:
3388:
3368:
3342:
2913:
2890:
2853:
2721:
2692:
2691:{\displaystyle y=f(x)}
2657:
2630:
2610:
2573:
2572:{\displaystyle y=f(x)}
2538:
2504:
2475:
2449:
2360:
2340:
2314:
2236:
2191:
2168:
2109:
2070:
2063:
2030:
1998:
1952:
1945:
1925:
1893:
1853:
1833:
1807:
1787:
1753:
1720:
1700:
1663:
1619:
1549:
1543:
1498:
1475:
1437:
1388:
1368:
1348:
1347:{\displaystyle y=mx+b}
1313:
1268:
1242:
1216:
1208:
1201:
1175:
1174:{\displaystyle y=f(x)}
1102:differential equations
843:Limit of distributions
663:Directional derivative
324:Faà di Bruno's formula
122:
24:
6564:Differential calculus
6525:Differential Calculus
6343:"Apollonius of Perga"
6333:Archimedes Palimpsest
6309:mathworld.wolfram.com
6278:Alcock, Lara (2016).
6155:
6108:
6088:
6046:
6004:
5984:
5292:
5112:
5029:
4991:
4965:
4923:
4865:
4775:
4685:Differential equation
4667:
4589:
4502:
4328:extreme value theorem
4311:first derivative test
4033:Augustin Louis Cauchy
3915:Treatise on Equations
3911:Sharaf al-Dīn al-Tūsī
3880:(c. 287–212 BC), and
3788:is a real number. If
3708:
3608:
3588:
3568:
3490:
3444:
3421:
3389:
3369:
3343:
2914:
2891:
2854:
2722:
2693:
2658:
2640:change. For example,
2631:
2611:
2574:
2539:
2537:{\displaystyle f'(x)}
2505:
2476:
2450:
2361:
2341:
2315:
2237:
2192:
2169:
2110:
2064:
2031:
2029:{\displaystyle (3,9)}
1999:
1997:{\displaystyle (2,4)}
1971:
1946:
1926:
1924:{\displaystyle (2,4)}
1894:
1865:
1854:
1834:
1808:
1788:
1754:
1721:
1701:
1664:
1620:
1544:
1510:
1499:
1476:
1438:
1414: change in
1389:
1369:
1349:
1314:
1269:
1243:
1214:
1202:
1176:
1145:
1118:differential geometry
1038:are connected by the
1021:graph of the function
978:differential calculus
927:Mathematical analysis
838:Generalized functions
523:arithmetico-geometric
369:Leibniz integral rule
123:
22:
6448:Sabra, A I. (1981).
6339:Robertson, Edmund F.
6127:
6097:
6055:
6013:
5993:
5951:
5872:invertible functions
5217:
5038:
5000:
4974:
4928:
4794:
4712:
4604:
4517:
4443:
4253:if it is zero, then
4072:Dirac delta function
4056:Lebesgue integration
3913:(1135–1213), in his
3715:polynomial functions
3625:
3597:
3577:
3499:
3453:
3430:
3398:
3378:
3355:
2926:
2900:
2867:
2734:
2720:{\displaystyle f(x)}
2702:
2667:
2644:
2620:
2583:
2548:
2514:
2503:{\displaystyle f(x)}
2485:
2462:
2373:
2350:
2327:
2249:
2201:
2178:
2119:
2078:
2040:
2008:
1976:
1956:The derivative of a
1935:
1903:
1870:
1843:
1817:
1797:
1764:
1730:
1710:
1673:
1669:is often written as
1632:
1556:
1515:
1485:
1447:
1443:. For, the graph of
1398:
1378:
1358:
1323:
1282:
1252:
1241:{\displaystyle f(x)}
1223:
1185:
1150:
1089:is a derivative. In
1029:linear approximation
1025:real-valued function
932:Nonstandard analysis
405:Lebesgue integration
275:Rules and identities
46:
6522:J. Edwards (1892).
6337:O'Connor, John J.;
6303:Weisstein, Eric W.
6093:. For this reason,
5731:. For instance, if
5729:graph of a function
4703:Newton's second law
4699:partial derivatives
4675:which is constant.
4305:This is called the
4250:is a local maximum;
4244:if it is negative,
4241:is a local minimum;
4235:if it is positive,
4060:absolute continuity
4026:theoretical physics
3909:The mathematician,
3882:Apollonius of Perga
3860:History of calculus
1832:{\displaystyle x=2}
1267:{\displaystyle x=a}
1200:{\displaystyle x=a}
1114:functional analysis
1091:operations research
603:Cauchy condensation
410:Contour integration
136:Fundamental theorem
63:
6546:2022-12-20 at the
6371:2016-09-01 at the
6366:Bhaskaracharya II.
6150:
6103:
6083:
6041:
5999:
5979:
5703:analytic functions
5287:
5210:. In other words,
5119:
5107:
5105:
5024:
4986:
4960:
4916:Mean value theorem
4910:Mean value theorem
4860:
4770:
4662:
4584:
4497:
4172:(and the value of
4010:Christiaan Huygens
3814:partial derivative
3703:
3603:
3583:
3563:
3485:
3442:{\displaystyle 2x}
3439:
3416:
3384:
3364:
3338:
3336:
3318:
3241:
3138:
3057:
2978:
2912:{\displaystyle 2x}
2909:
2886:
2849:
2798:
2717:
2688:
2656:{\displaystyle dx}
2653:
2626:
2606:
2569:
2534:
2500:
2471:
2445:
2394:
2356:
2336:
2310:
2232:
2187:
2164:
2105:
2071:
2059:
2026:
1994:
1953:
1941:
1921:
1889:
1849:
1829:
1803:
1783:
1749:
1716:
1696:
1659:
1615:
1550:
1539:
1497:{\displaystyle -2}
1494:
1471:
1433:
1384:
1364:
1344:
1309:
1278:of the tangent to
1264:
1238:
1219:The derivative of
1217:
1209:
1197:
1171:
775:Partial derivative
704:generalized Stokes
598:Alternating series
479:Reduction formulae
454:tangent half-angle
441:Cylindrical shells
364:Integral transform
359:Lists of integrals
163:Mean value theorem
118:
49:
25:
6329:Euclid's Elements
6289:978-0-19-872353-0
6171:differentiability
6148:
6106:{\displaystyle x}
6002:{\displaystyle x}
5874:pasted together.
5640:Taylor polynomial
5610:should always be
5576:should always be
5329:Taylor polynomial
5282:
5144:are numbers with
5104:
4855:
4815:
4765:
4616:
4529:
4417:Newtonian physics
4355:second derivative
4343:higher dimensions
4217:second derivative
4169:stationary points
4076:Quantum Mechanics
4039:(1826–1866), and
3888:also made use of
3884:(c. 262–190 BC).
3731:differentiability
3641:
3606:{\displaystyle n}
3586:{\displaystyle a}
3536:
3474:
3387:{\displaystyle 0}
3300:
3288:
3223:
3211:
3120:
3108:
3039:
3027:
2960:
2951:
2847:
2780:
2755:
2629:{\displaystyle d}
2604:
2443:
2376:
2359:{\displaystyle 0}
2308:
2255:
2230:
2207:
1944:{\displaystyle 4}
1852:{\displaystyle 4}
1806:{\displaystyle 4}
1694:
1657:
1651:
1641:
1604:
1581:
1575:
1565:
1431:
1425:
1415:
1404:
1387:{\displaystyle x}
1374:by the change in
1367:{\displaystyle y}
1106:natural phenomena
1098:maxima and minima
1087:chemical reaction
1036:integral calculus
986:integral calculus
980:is a subfield of
970:
969:
850:
849:
812:
811:
780:Multiple integral
716:
715:
620:
619:
587:Direct comparison
558:Convergence tests
496:
495:
469:Partial fractions
336:
335:
246:Second derivative
6576:
6533:
6511:
6473:
6472:Eves, H. (1990).
6470:
6464:
6463:
6445:
6439:
6433:
6422:
6421:
6381:
6375:
6362:
6356:
6355:
6325:
6319:
6318:
6316:
6315:
6300:
6294:
6293:
6275:
6269:
6268:
6266:
6265:
6251:
6245:
6244:
6242:
6241:
6227:
6205:
6202:Pierre de Fermat
6194:
6188:
6180:
6174:
6167:
6161:
6159:
6157:
6156:
6151:
6149:
6147:
6139:
6131:
6120:
6114:
6112:
6110:
6109:
6104:
6092:
6090:
6089:
6084:
6082:
6081:
6050:
6048:
6047:
6042:
6040:
6039:
6008:
6006:
6005:
6000:
5988:
5986:
5985:
5980:
5978:
5977:
5941:
5935:
5932:
5914:
5913:
5862:
5858:
5854:
5850:
5846:
5845:
5844:
5833:
5827:
5817:
5811:
5792:
5786:
5780:
5765:
5753:
5707:smooth functions
5693:
5687:
5681:
5675:
5668:Taylor's theorem
5665:
5659:
5653:
5647:
5637:
5636:
5634:
5633:
5630:
5627:
5609:
5603:
5602:
5600:
5599:
5596:
5593:
5575:
5569:
5563:
5557:
5551:
5537:
5531:
5517:
5511:
5495:
5489:
5483:
5477:
5471:
5419:
5382:
5357:
5348:
5318:
5312:
5305:
5296:
5294:
5293:
5288:
5283:
5281:
5270:
5241:
5227:
5209:
5203:
5197:
5191:
5185:
5169:
5153:
5143:
5137:
5131:
5116:
5114:
5113:
5108:
5106:
5103:
5092:
5063:
5048:
5033:
5031:
5030:
5025:
4995:
4993:
4992:
4987:
4969:
4967:
4966:
4961:
4959:
4905:
4899:
4893:
4887:
4869:
4867:
4866:
4861:
4856:
4854:
4853:
4852:
4839:
4835:
4834:
4824:
4816:
4814:
4806:
4798:
4779:
4777:
4776:
4771:
4766:
4764:
4763:
4762:
4749:
4745:
4744:
4734:
4671:
4669:
4668:
4663:
4638:
4618:
4617:
4609:
4593:
4591:
4590:
4585:
4551:
4531:
4530:
4522:
4506:
4504:
4503:
4498:
4476:
4475:
4318:
4300:
4293:
4279:
4272:
4258:
4249:
4240:
4230:
4224:
4214:
4208:
4202:
4193:
4183:
4177:
4159:
4148:
4142:
4136:
4122:
4112:
4104:
4068:Laurent Schwartz
4041:Karl Weierstrass
4037:Bernhard Riemann
4016:(1623–1662) and
3998:Pierre de Fermat
3987:(1643–1727) and
3975:
3963:
3952:
3941:
3930:
3920:
3848:total derivative
3845:
3839:
3838:
3836:
3835:
3829:
3826:
3811:
3805:
3799:
3793:
3787:
3781:
3775:
3769:
3763:
3757:
3751:
3745:
3712:
3710:
3709:
3704:
3702:
3701:
3686:
3685:
3658:
3657:
3642:
3640:
3629:
3612:
3610:
3609:
3604:
3592:
3590:
3589:
3584:
3572:
3570:
3569:
3564:
3562:
3561:
3537:
3535:
3527:
3523:
3522:
3503:
3494:
3492:
3491:
3486:
3475:
3473:
3465:
3457:
3448:
3446:
3445:
3440:
3425:
3423:
3422:
3417:
3393:
3391:
3390:
3385:
3373:
3371:
3370:
3365:
3347:
3345:
3344:
3339:
3337:
3317:
3293:
3289:
3287:
3279:
3278:
3277:
3243:
3240:
3216:
3212:
3210:
3202:
3201:
3200:
3188:
3187:
3151:
3150:
3140:
3137:
3113:
3109:
3107:
3099:
3098:
3097:
3085:
3084:
3059:
3056:
3032:
3028:
3026:
3018:
2980:
2977:
2952:
2950:
2942:
2934:
2918:
2916:
2915:
2910:
2895:
2893:
2892:
2887:
2885:
2884:
2858:
2856:
2855:
2850:
2848:
2846:
2838:
2800:
2797:
2767:
2756:
2754:
2746:
2738:
2726:
2724:
2723:
2718:
2697:
2695:
2694:
2689:
2662:
2660:
2659:
2654:
2636:representing an
2635:
2633:
2632:
2627:
2615:
2613:
2612:
2607:
2605:
2603:
2595:
2587:
2578:
2576:
2575:
2570:
2543:
2541:
2540:
2535:
2524:
2509:
2507:
2506:
2501:
2480:
2478:
2477:
2472:
2454:
2452:
2451:
2446:
2444:
2442:
2434:
2396:
2393:
2365:
2363:
2362:
2357:
2345:
2343:
2342:
2337:
2319:
2317:
2316:
2311:
2309:
2307:
2299:
2261:
2256:
2253:
2241:
2239:
2238:
2233:
2231:
2229:
2221:
2213:
2208:
2205:
2196:
2194:
2193:
2188:
2173:
2171:
2170:
2165:
2114:
2112:
2111:
2106:
2068:
2066:
2065:
2060:
2058:
2057:
2035:
2033:
2032:
2027:
2003:
2001:
2000:
1995:
1950:
1948:
1947:
1942:
1930:
1928:
1927:
1922:
1898:
1896:
1895:
1890:
1888:
1887:
1858:
1856:
1855:
1850:
1838:
1836:
1835:
1830:
1812:
1810:
1809:
1804:
1792:
1790:
1789:
1784:
1782:
1781:
1758:
1756:
1755:
1750:
1748:
1747:
1725:
1723:
1722:
1717:
1705:
1703:
1702:
1697:
1695:
1693:
1685:
1677:
1668:
1666:
1665:
1660:
1658:
1656:
1652:
1649:
1646:
1642:
1639:
1636:
1624:
1622:
1621:
1616:
1605:
1603:
1595:
1587:
1582:
1580:
1576:
1573:
1570:
1566:
1563:
1560:
1548:
1546:
1545:
1540:
1503:
1501:
1500:
1495:
1480:
1478:
1477:
1472:
1442:
1440:
1439:
1434:
1432:
1430:
1426:
1423:
1420:
1416:
1413:
1410:
1405:
1402:
1393:
1391:
1390:
1385:
1373:
1371:
1370:
1365:
1353:
1351:
1350:
1345:
1318:
1316:
1315:
1310:
1273:
1271:
1270:
1265:
1247:
1245:
1244:
1239:
1206:
1204:
1203:
1198:
1180:
1178:
1177:
1172:
1126:abstract algebra
1110:complex analysis
1076:
962:
955:
948:
896:
861:
827:
826:
823:
790:Surface integral
733:
732:
729:
637:
636:
633:
593:Limit comparison
513:
512:
509:
400:Riemann integral
353:
352:
349:
309:L'Hôpital's rule
266:Taylor's theorem
187:
186:
183:
127:
125:
124:
119:
71:
62:
57:
27:
26:
6584:
6583:
6579:
6578:
6577:
6575:
6574:
6573:
6554:
6553:
6548:Wayback Machine
6518:
6482:
6477:
6476:
6471:
6467:
6460:
6446:
6442:
6434:
6425:
6402:10.2307/3614212
6382:
6378:
6373:Wayback Machine
6364:Ian G. Pearce.
6363:
6359:
6326:
6322:
6313:
6311:
6301:
6297:
6290:
6276:
6272:
6263:
6261:
6253:
6252:
6248:
6239:
6237:
6229:
6228:
6224:
6219:
6214:
6209:
6208:
6195:
6191:
6181:
6177:
6168:
6164:
6140:
6132:
6130:
6128:
6125:
6124:
6121:
6117:
6098:
6095:
6094:
6077:
6073:
6056:
6053:
6052:
6035:
6031:
6014:
6011:
6010:
5994:
5991:
5990:
5973:
5969:
5952:
5949:
5948:
5942:
5938:
5933:
5929:
5924:
5908:
5880:
5860:
5856:
5852:
5848:
5839:
5837:
5835:
5829:
5823:
5813:
5798:
5788:
5782:
5767:
5755:
5732:
5721:
5715:
5689:
5683:
5677:
5671:
5661:
5655:
5649:
5643:
5631:
5628:
5625:
5615:
5614:
5612:
5611:
5605:
5597:
5594:
5591:
5581:
5580:
5578:
5577:
5571:
5565:
5559:
5553:
5549:
5539:
5533:
5529:
5519:
5513:
5510:
5504:
5491:
5485:
5479:
5473:
5469:
5454:
5439:
5421:
5417:
5402:
5384:
5380:
5362:
5356:
5350:
5339:
5335:
5327:Main articles:
5325:
5314:
5308:
5301:
5271:
5242:
5240:
5220:
5218:
5215:
5214:
5205:
5199:
5193:
5187:
5171:
5155:
5145:
5139:
5133:
5122:
5093:
5064:
5061:
5041:
5039:
5036:
5035:
5001:
4998:
4997:
4975:
4972:
4971:
4955:
4929:
4926:
4925:
4918:
4912:
4901:
4895:
4889:
4874:
4848:
4844:
4840:
4830:
4826:
4825:
4823:
4807:
4799:
4797:
4795:
4792:
4791:
4758:
4754:
4750:
4740:
4736:
4735:
4733:
4713:
4710:
4709:
4687:
4681:
4631:
4608:
4607:
4605:
4602:
4601:
4544:
4521:
4520:
4518:
4515:
4514:
4471:
4467:
4444:
4441:
4440:
4412:time derivative
4403:
4395:minimal surface
4382:
4376:
4332:closed interval
4314:
4295:
4281:
4274:
4260:
4254:
4245:
4236:
4226:
4220:
4210:
4204:
4198:
4189:
4179:
4173:
4163:critical points
4150:
4144:
4138:
4132:
4118:
4110:
4100:
4097:
4092:
4045:Euclidean space
3965:
3954:
3943:
3932:
3922:
3921:) of the cubic
3918:
3904:Rolle's theorem
3862:
3856:
3841:
3830:
3827:
3821:
3820:
3818:
3817:
3807:
3806:depends on how
3801:
3795:
3789:
3783:
3777:
3771:
3765:
3759:
3753:
3747:
3741:
3697:
3693:
3681:
3677:
3653:
3649:
3633:
3628:
3626:
3623:
3622:
3621:. For example,
3598:
3595:
3594:
3578:
3575:
3574:
3551:
3547:
3528:
3518:
3514:
3504:
3502:
3500:
3497:
3496:
3466:
3458:
3456:
3454:
3451:
3450:
3431:
3428:
3427:
3399:
3396:
3395:
3379:
3376:
3375:
3356:
3353:
3352:
3335:
3334:
3304:
3291:
3290:
3280:
3273:
3269:
3244:
3242:
3227:
3214:
3213:
3203:
3196:
3192:
3183:
3179:
3146:
3142:
3141:
3139:
3124:
3111:
3110:
3100:
3093:
3089:
3080:
3076:
3060:
3058:
3043:
3030:
3029:
3019:
2981:
2979:
2964:
2953:
2943:
2935:
2933:
2929:
2927:
2924:
2923:
2901:
2898:
2897:
2880:
2876:
2868:
2865:
2864:
2839:
2801:
2799:
2784:
2760:
2747:
2739:
2737:
2735:
2732:
2731:
2703:
2700:
2699:
2668:
2665:
2664:
2645:
2642:
2641:
2621:
2618:
2617:
2596:
2588:
2586:
2584:
2581:
2580:
2549:
2546:
2545:
2517:
2515:
2512:
2511:
2486:
2483:
2482:
2463:
2460:
2459:
2435:
2397:
2395:
2380:
2374:
2371:
2370:
2351:
2348:
2347:
2328:
2325:
2324:
2300:
2262:
2260:
2252:
2250:
2247:
2246:
2222:
2214:
2212:
2204:
2202:
2199:
2198:
2179:
2176:
2175:
2120:
2117:
2116:
2079:
2076:
2075:
2073:
2053:
2049:
2041:
2038:
2037:
2009:
2006:
2005:
1977:
1974:
1973:
1955:
1936:
1933:
1932:
1904:
1901:
1900:
1883:
1879:
1871:
1868:
1867:
1844:
1841:
1840:
1818:
1815:
1814:
1798:
1795:
1794:
1793:has a slope of
1777:
1773:
1765:
1762:
1761:
1743:
1739:
1731:
1728:
1727:
1711:
1708:
1707:
1686:
1678:
1676:
1674:
1671:
1670:
1650:change in
1648:
1647:
1640:change in
1638:
1637:
1635:
1633:
1630:
1629:
1596:
1588:
1586:
1574:change in
1572:
1571:
1564:change in
1562:
1561:
1559:
1557:
1554:
1553:
1516:
1513:
1512:
1486:
1483:
1482:
1481:has a slope of
1448:
1445:
1444:
1424:change in
1422:
1421:
1412:
1411:
1409:
1401:
1399:
1396:
1395:
1394:, meaning that
1379:
1376:
1375:
1359:
1356:
1355:
1324:
1321:
1320:
1283:
1280:
1279:
1253:
1250:
1249:
1224:
1221:
1220:
1186:
1183:
1182:
1151:
1148:
1147:
1140:
1134:
1074:
1009:differentiation
966:
937:
936:
922:Integration Bee
897:
894:
887:
886:
862:
859:
852:
851:
824:
821:
814:
813:
795:Volume integral
730:
725:
718:
717:
634:
629:
622:
621:
591:
510:
505:
498:
497:
489:Risch algorithm
464:Euler's formula
350:
345:
338:
337:
319:General Leibniz
202:generalizations
184:
179:
172:
158:Rolle's theorem
153:
128:
64:
58:
53:
47:
44:
43:
17:
12:
11:
5:
6582:
6572:
6571:
6566:
6552:
6551:
6534:
6517:
6514:
6513:
6512:
6500:10.2307/604533
6494:(2): 304–309.
6481:
6478:
6475:
6474:
6465:
6459:978-0521284363
6458:
6440:
6423:
6396:(381): 307–8.
6376:
6357:
6320:
6295:
6288:
6270:
6246:
6221:
6220:
6218:
6215:
6213:
6210:
6207:
6206:
6189:
6175:
6162:
6146:
6143:
6138:
6135:
6115:
6102:
6080:
6076:
6072:
6069:
6066:
6063:
6060:
6038:
6034:
6030:
6027:
6024:
6021:
6018:
5998:
5976:
5972:
5968:
5965:
5962:
5959:
5956:
5936:
5926:
5925:
5923:
5920:
5919:
5918:
5906:
5901:
5896:
5891:
5886:
5879:
5876:
5717:Main article:
5714:
5711:
5623:
5589:
5547:
5527:
5508:
5467:
5452:
5437:
5415:
5400:
5378:
5354:
5324:
5321:
5298:
5297:
5286:
5280:
5277:
5274:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5239:
5236:
5233:
5230:
5226:
5223:
5192:at some point
5102:
5099:
5096:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5060:
5057:
5054:
5051:
5047:
5044:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
4985:
4982:
4979:
4958:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4914:Main article:
4911:
4908:
4871:
4870:
4859:
4851:
4847:
4843:
4838:
4833:
4829:
4822:
4819:
4813:
4810:
4805:
4802:
4781:
4780:
4769:
4761:
4757:
4753:
4748:
4743:
4739:
4732:
4729:
4726:
4723:
4720:
4717:
4683:Main article:
4680:
4677:
4673:
4672:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4637:
4634:
4630:
4627:
4624:
4621:
4615:
4612:
4595:
4594:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4550:
4547:
4543:
4540:
4537:
4534:
4528:
4525:
4508:
4507:
4494:
4491:
4488:
4485:
4482:
4479:
4474:
4470:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4434:
4433:
4427:
4402:
4399:
4378:Main article:
4375:
4372:
4363:Hessian matrix
4303:
4302:
4251:
4242:
4186:critical value
4096:
4093:
4091:
4088:
4080:weak solutions
4006:René Descartes
3858:Main article:
3855:
3852:
3700:
3696:
3692:
3689:
3684:
3680:
3676:
3673:
3670:
3667:
3664:
3661:
3656:
3652:
3648:
3645:
3639:
3636:
3632:
3602:
3582:
3560:
3557:
3554:
3550:
3546:
3543:
3540:
3534:
3531:
3526:
3521:
3517:
3513:
3510:
3507:
3484:
3481:
3478:
3472:
3469:
3464:
3461:
3438:
3435:
3415:
3412:
3409:
3406:
3403:
3383:
3363:
3360:
3349:
3348:
3333:
3330:
3327:
3324:
3321:
3316:
3313:
3310:
3307:
3303:
3299:
3296:
3294:
3292:
3286:
3283:
3276:
3272:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3239:
3236:
3233:
3230:
3226:
3222:
3219:
3217:
3215:
3209:
3206:
3199:
3195:
3191:
3186:
3182:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3149:
3145:
3136:
3133:
3130:
3127:
3123:
3119:
3116:
3114:
3112:
3106:
3103:
3096:
3092:
3088:
3083:
3079:
3075:
3072:
3069:
3066:
3063:
3055:
3052:
3049:
3046:
3042:
3038:
3035:
3033:
3031:
3025:
3022:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2976:
2973:
2970:
2967:
2963:
2959:
2956:
2954:
2949:
2946:
2941:
2938:
2932:
2931:
2908:
2905:
2883:
2879:
2875:
2872:
2860:
2859:
2845:
2842:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2796:
2793:
2790:
2787:
2783:
2779:
2776:
2773:
2770:
2766:
2763:
2759:
2753:
2750:
2745:
2742:
2716:
2713:
2710:
2707:
2687:
2684:
2681:
2678:
2675:
2672:
2652:
2649:
2625:
2602:
2599:
2594:
2591:
2568:
2565:
2562:
2559:
2556:
2553:
2533:
2530:
2527:
2523:
2520:
2499:
2496:
2493:
2490:
2470:
2467:
2456:
2455:
2441:
2438:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2392:
2389:
2386:
2383:
2379:
2355:
2335:
2332:
2321:
2320:
2306:
2303:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2259:
2228:
2225:
2220:
2217:
2211:
2186:
2183:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2056:
2052:
2048:
2045:
2025:
2022:
2019:
2016:
2013:
1993:
1990:
1987:
1984:
1981:
1940:
1920:
1917:
1914:
1911:
1908:
1886:
1882:
1878:
1875:
1848:
1828:
1825:
1822:
1802:
1780:
1776:
1772:
1769:
1746:
1742:
1738:
1735:
1715:
1692:
1689:
1684:
1681:
1655:
1645:
1626:
1625:
1614:
1611:
1608:
1602:
1599:
1594:
1591:
1585:
1579:
1569:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1493:
1490:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1429:
1419:
1408:
1383:
1363:
1343:
1340:
1337:
1334:
1331:
1328:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1263:
1260:
1257:
1237:
1234:
1231:
1228:
1196:
1193:
1190:
1170:
1167:
1164:
1161:
1158:
1155:
1136:Main article:
1133:
1130:
1122:measure theory
1005:rate of change
968:
967:
965:
964:
957:
950:
942:
939:
938:
935:
934:
929:
924:
919:
917:List of topics
914:
909:
904:
898:
893:
892:
889:
888:
885:
884:
879:
874:
869:
863:
858:
857:
854:
853:
848:
847:
846:
845:
840:
835:
825:
820:
819:
816:
815:
810:
809:
808:
807:
802:
797:
792:
787:
782:
777:
769:
768:
764:
763:
762:
761:
756:
751:
746:
738:
737:
731:
724:
723:
720:
719:
714:
713:
712:
711:
706:
701:
696:
691:
686:
678:
677:
673:
672:
671:
670:
665:
660:
655:
650:
645:
635:
628:
627:
624:
623:
618:
617:
616:
615:
610:
605:
600:
595:
589:
584:
579:
574:
569:
561:
560:
554:
553:
552:
551:
546:
541:
536:
531:
526:
511:
504:
503:
500:
499:
494:
493:
492:
491:
486:
481:
476:
474:Changing order
471:
466:
461:
443:
438:
433:
425:
424:
423:Integration by
420:
419:
418:
417:
412:
407:
402:
397:
387:
385:Antiderivative
379:
378:
374:
373:
372:
371:
366:
361:
351:
344:
343:
340:
339:
334:
333:
332:
331:
326:
321:
316:
311:
306:
301:
296:
291:
286:
278:
277:
271:
270:
269:
268:
263:
258:
253:
248:
243:
235:
234:
230:
229:
228:
227:
226:
225:
220:
215:
205:
192:
191:
185:
178:
177:
174:
173:
171:
170:
165:
160:
154:
152:
151:
146:
140:
139:
138:
130:
129:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
80:
77:
74:
70:
67:
61:
56:
52:
42:
39:
38:
32:
31:
15:
9:
6:
4:
3:
2:
6581:
6570:
6567:
6565:
6562:
6561:
6559:
6549:
6545:
6542:
6539:
6535:
6531:
6527:
6526:
6520:
6519:
6516:Other sources
6509:
6505:
6501:
6497:
6493:
6489:
6484:
6483:
6469:
6461:
6455:
6451:
6444:
6437:
6436:Berggren 1990
6432:
6430:
6428:
6419:
6415:
6411:
6407:
6403:
6399:
6395:
6391:
6387:
6380:
6374:
6370:
6367:
6361:
6354:
6350:
6349:
6344:
6340:
6334:
6330:
6324:
6310:
6306:
6299:
6291:
6285:
6281:
6274:
6260:
6256:
6250:
6236:
6232:
6226:
6222:
6203:
6199:
6198:James Gregory
6193:
6186:
6179:
6172:
6166:
6144:
6141:
6136:
6133:
6119:
6100:
6078:
6074:
6070:
6064:
6058:
6036:
6032:
6028:
6022:
6016:
5996:
5974:
5970:
5966:
5960:
5954:
5946:
5940:
5931:
5927:
5917:
5912:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5881:
5875:
5873:
5869:
5864:
5843:
5832:
5826:
5821:
5816:
5809:
5805:
5801:
5796:
5793:, which is a
5791:
5785:
5778:
5774:
5770:
5763:
5759:
5751:
5747:
5743:
5739:
5735:
5730:
5726:
5720:
5710:
5708:
5704:
5700:
5699:Taylor series
5695:
5692:
5686:
5680:
5674:
5669:
5664:
5658:
5652:
5646:
5641:
5622:
5618:
5608:
5588:
5584:
5574:
5568:
5562:
5556:
5546:
5542:
5536:
5526:
5522:
5516:
5507:
5502:
5501:neighbourhood
5497:
5494:
5488:
5482:
5476:
5466:
5462:
5458:
5451:
5447:
5443:
5436:
5432:
5428:
5424:
5414:
5410:
5406:
5399:
5395:
5391:
5387:
5377:
5373:
5369:
5365:
5361:
5353:
5349:at the point
5346:
5342:
5334:
5333:Taylor series
5330:
5320:
5317:
5311:
5304:
5284:
5278:
5275:
5272:
5264:
5258:
5255:
5249:
5243:
5237:
5231:
5224:
5221:
5213:
5212:
5211:
5208:
5202:
5196:
5190:
5183:
5179:
5175:
5167:
5163:
5159:
5152:
5148:
5142:
5136:
5129:
5125:
5100:
5097:
5094:
5086:
5080:
5077:
5071:
5065:
5058:
5052:
5045:
5042:
5018:
5015:
5012:
5006:
5003:
4983:
4980:
4977:
4946:
4943:
4940:
4934:
4931:
4922:
4917:
4907:
4904:
4898:
4892:
4885:
4881:
4877:
4857:
4849:
4845:
4836:
4831:
4820:
4817:
4811:
4803:
4790:
4789:
4788:
4786:
4785:heat equation
4767:
4759:
4755:
4751:
4746:
4741:
4737:
4730:
4727:
4721:
4715:
4708:
4707:
4706:
4704:
4700:
4696:
4692:
4686:
4676:
4657:
4654:
4651:
4648:
4642:
4635:
4632:
4628:
4622:
4613:
4610:
4600:
4599:
4598:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4555:
4548:
4545:
4541:
4535:
4526:
4523:
4513:
4512:
4511:
4492:
4489:
4486:
4483:
4480:
4477:
4472:
4468:
4464:
4461:
4458:
4452:
4446:
4439:
4438:
4437:
4431:
4428:
4425:
4422:
4421:
4420:
4418:
4414:
4413:
4408:
4398:
4396:
4392:
4388:
4387:shortest path
4381:
4371:
4369:
4364:
4360:
4356:
4353:is zero. The
4352:
4348:
4347:scalar valued
4344:
4339:
4335:
4333:
4329:
4325:
4320:
4317:
4312:
4308:
4298:
4292:
4288:
4284:
4277:
4271:
4267:
4263:
4257:
4252:
4248:
4243:
4239:
4234:
4233:
4232:
4229:
4223:
4218:
4213:
4207:
4201:
4195:
4192:
4187:
4182:
4176:
4171:
4170:
4165:
4164:
4157:
4153:
4147:
4141:
4135:
4130:
4129:local minimum
4126:
4125:local maximum
4121:
4116:
4115:open interval
4108:
4103:
4087:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4052:
4050:
4049:complex plane
4046:
4042:
4038:
4035:(1789–1857),
4034:
4029:
4027:
4023:
4019:
4015:
4014:Blaise Pascal
4012:(1629–1695),
4011:
4008:(1596–1650),
4007:
4004:(1630–1677),
4003:
4000:(1607-1665),
3999:
3994:
3990:
3986:
3981:
3979:
3978:Roshdi Rashed
3973:
3969:
3961:
3957:
3950:
3946:
3939:
3935:
3929:
3925:
3916:
3912:
3907:
3905:
3901:
3897:
3896:
3891:
3887:
3883:
3879:
3876:(c. 300 BC),
3875:
3871:
3867:
3861:
3851:
3849:
3844:
3834:
3825:
3815:
3810:
3804:
3798:
3792:
3786:
3780:
3774:
3768:
3762:
3756:
3750:
3744:
3739:
3734:
3732:
3728:
3727:quotient rule
3724:
3720:
3716:
3698:
3694:
3690:
3687:
3682:
3678:
3671:
3665:
3662:
3654:
3650:
3646:
3637:
3634:
3630:
3620:
3616:
3600:
3580:
3558:
3555:
3552:
3548:
3544:
3541:
3538:
3532:
3529:
3519:
3515:
3511:
3505:
3482:
3479:
3476:
3470:
3467:
3462:
3459:
3449:. Therefore,
3436:
3433:
3413:
3407:
3404:
3401:
3381:
3361:
3331:
3325:
3322:
3319:
3314:
3308:
3297:
3295:
3284:
3274:
3266:
3257:
3254:
3248:
3245:
3237:
3231:
3220:
3218:
3207:
3197:
3193:
3189:
3184:
3176:
3167:
3164:
3158:
3155:
3152:
3147:
3143:
3134:
3128:
3117:
3115:
3104:
3094:
3090:
3086:
3081:
3073:
3067:
3064:
3053:
3047:
3036:
3034:
3023:
3012:
3006:
3003:
2997:
2991:
2988:
2982:
2974:
2968:
2957:
2955:
2947:
2944:
2939:
2936:
2922:
2921:
2920:
2906:
2903:
2881:
2877:
2873:
2870:
2843:
2832:
2826:
2823:
2817:
2811:
2808:
2802:
2794:
2788:
2777:
2771:
2764:
2761:
2757:
2751:
2748:
2743:
2740:
2730:
2729:
2728:
2711:
2705:
2682:
2676:
2673:
2670:
2650:
2647:
2639:
2638:infinitesimal
2623:
2600:
2597:
2592:
2589:
2563:
2557:
2554:
2551:
2528:
2521:
2518:
2494:
2488:
2468:
2439:
2428:
2422:
2419:
2413:
2407:
2404:
2398:
2390:
2384:
2369:
2368:
2367:
2353:
2333:
2304:
2293:
2287:
2284:
2278:
2272:
2269:
2263:
2257:
2245:
2244:
2243:
2242:. This gives
2226:
2218:
2209:
2184:
2155:
2149:
2146:
2140:
2137:
2134:
2128:
2125:
2096:
2090:
2087:
2084:
2054:
2050:
2046:
2043:
2020:
2017:
2014:
1988:
1985:
1982:
1970:
1966:
1964:
1959:
1938:
1915:
1912:
1909:
1884:
1880:
1876:
1873:
1866:The graph of
1864:
1860:
1846:
1826:
1823:
1820:
1800:
1778:
1774:
1770:
1767:
1744:
1740:
1736:
1733:
1690:
1682:
1653:
1643:
1628:For brevity,
1612:
1609:
1606:
1600:
1597:
1592:
1589:
1583:
1577:
1567:
1552:
1551:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1511:The graph of
1509:
1505:
1491:
1488:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1427:
1417:
1406:
1381:
1361:
1341:
1338:
1335:
1332:
1329:
1326:
1300:
1294:
1291:
1288:
1277:
1261:
1258:
1255:
1248:at the point
1232:
1226:
1213:
1194:
1191:
1188:
1165:
1159:
1156:
1153:
1144:
1139:
1129:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1094:
1092:
1088:
1084:
1083:reaction rate
1080:
1072:
1068:
1064:
1060:
1056:
1052:
1047:
1045:
1041:
1037:
1032:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
989:
987:
983:
979:
975:
963:
958:
956:
951:
949:
944:
943:
941:
940:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
899:
891:
890:
883:
880:
878:
875:
873:
870:
868:
865:
864:
856:
855:
844:
841:
839:
836:
834:
831:
830:
829:
828:
818:
817:
806:
803:
801:
798:
796:
793:
791:
788:
786:
785:Line integral
783:
781:
778:
776:
773:
772:
771:
770:
766:
765:
760:
757:
755:
752:
750:
747:
745:
742:
741:
740:
739:
735:
734:
728:
727:Multivariable
722:
721:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
681:
680:
679:
675:
674:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
640:
639:
638:
632:
626:
625:
614:
611:
609:
606:
604:
601:
599:
596:
594:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
564:
563:
562:
559:
556:
555:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
524:
520:
517:
516:
515:
514:
508:
502:
501:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
459:
455:
451:
450:trigonometric
447:
444:
442:
439:
437:
434:
432:
429:
428:
427:
426:
422:
421:
416:
413:
411:
408:
406:
403:
401:
398:
395:
391:
388:
386:
383:
382:
381:
380:
376:
375:
370:
367:
365:
362:
360:
357:
356:
355:
354:
348:
342:
341:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
281:
280:
279:
276:
273:
272:
267:
264:
262:
261:Related rates
259:
257:
254:
252:
249:
247:
244:
242:
239:
238:
237:
236:
232:
231:
224:
221:
219:
218:of a function
216:
214:
213:infinitesimal
211:
210:
209:
206:
203:
199:
196:
195:
194:
193:
189:
188:
182:
176:
175:
169:
166:
164:
161:
159:
156:
155:
150:
147:
145:
142:
141:
137:
134:
133:
132:
131:
112:
106:
103:
97:
91:
88:
85:
82:
75:
68:
65:
59:
54:
50:
41:
40:
37:
34:
33:
29:
28:
21:
6537:
6524:
6491:
6487:
6468:
6449:
6443:
6393:
6389:
6385:
6379:
6360:
6346:
6323:
6312:. Retrieved
6308:
6305:"Derivative"
6298:
6279:
6273:
6262:. Retrieved
6258:
6249:
6238:. Retrieved
6234:
6225:
6204:(1601–1665).
6192:
6178:
6165:
6118:
6051:and writing
5939:
5930:
5865:
5841:
5830:
5824:
5814:
5807:
5803:
5799:
5789:
5783:
5776:
5772:
5768:
5761:
5757:
5749:
5745:
5741:
5737:
5733:
5722:
5698:
5696:
5690:
5684:
5678:
5672:
5662:
5656:
5650:
5644:
5639:
5620:
5616:
5606:
5586:
5582:
5572:
5566:
5560:
5554:
5544:
5540:
5534:
5524:
5520:
5514:
5505:
5498:
5492:
5486:
5480:
5474:
5464:
5460:
5456:
5449:
5445:
5441:
5434:
5430:
5426:
5422:
5412:
5408:
5404:
5397:
5393:
5389:
5385:
5375:
5371:
5367:
5363:
5358:is a linear
5351:
5344:
5340:
5336:
5315:
5309:
5302:
5299:
5206:
5200:
5194:
5188:
5181:
5177:
5173:
5165:
5161:
5157:
5150:
5146:
5140:
5134:
5127:
5123:
5120:
4902:
4896:
4890:
4883:
4879:
4875:
4872:
4782:
4688:
4674:
4596:
4509:
4435:
4430:acceleration
4410:
4404:
4383:
4368:saddle point
4340:
4336:
4324:optimization
4321:
4315:
4304:
4296:
4290:
4286:
4282:
4275:
4269:
4265:
4261:
4255:
4246:
4237:
4227:
4221:
4211:
4205:
4199:
4196:
4190:
4185:
4184:is called a
4180:
4174:
4167:
4161:
4155:
4151:
4145:
4139:
4133:
4119:
4101:
4098:
4095:Optimization
4062:. Later the
4053:
4030:
4021:
4002:Isaac Barrow
3985:Isaac Newton
3982:
3971:
3967:
3959:
3955:
3948:
3944:
3937:
3933:
3931:occurs when
3927:
3923:
3914:
3908:
3893:
3890:indivisibles
3866:tangent line
3863:
3842:
3832:
3823:
3808:
3802:
3796:
3790:
3784:
3778:
3772:
3766:
3760:
3754:
3748:
3742:
3738:differential
3735:
3723:product rule
3350:
2861:
2457:
2322:
2072:
1954:
1627:
1218:
1095:
1063:acceleration
1055:displacement
1048:
1033:
1017:tangent line
1008:
1001:differential
990:
977:
971:
446:Substitution
208:Differential
181:Differential
180:
6480:Works cited
4996:there is a
4359:eigenvalues
4160:are called
4018:John Wallis
3900:Bhāskara II
3426:approaches
3374:approaches
2206:slope
1963:secant line
1403:slope
1044:integration
974:mathematics
902:Precalculus
895:Miscellanea
860:Specialized
767:Definitions
534:Alternating
377:Definitions
190:Definitions
6558:Categories
6314:2020-07-26
6264:2020-05-09
6240:2020-05-09
6212:References
5795:paraboloid
5766:such that
5532:, and for
5360:polynomial
3886:Archimedes
3878:Archimedes
3719:chain rule
3619:power rule
1138:Derivative
1132:Derivative
993:derivative
882:Variations
877:Stochastic
867:Fractional
736:Formalisms
699:Divergence
668:Identities
648:Divergence
198:Derivative
149:Continuity
6418:176660647
6217:Citations
5276:−
5256:−
5098:−
5078:−
5007:∈
4953:→
4894:and time
4842:∂
4828:∂
4821:α
4809:∂
4801:∂
4652:−
4614:¨
4565:−
4527:˙
4462:−
4391:geodesics
4326:. By the
3615:constants
3556:−
3411:Δ
3359:Δ
3329:Δ
3312:→
3306:Δ
3282:Δ
3264:Δ
3252:Δ
3235:→
3229:Δ
3205:Δ
3190:−
3174:Δ
3162:Δ
3132:→
3126:Δ
3102:Δ
3087:−
3071:Δ
3051:→
3045:Δ
3021:Δ
3004:−
2995:Δ
2972:→
2966:Δ
2841:Δ
2824:−
2815:Δ
2792:→
2786:Δ
2466:Δ
2437:Δ
2420:−
2411:Δ
2388:→
2382:Δ
2331:Δ
2302:Δ
2285:−
2276:Δ
2224:Δ
2216:Δ
2182:Δ
2153:Δ
2132:Δ
1714:Δ
1688:Δ
1680:Δ
1610:−
1590:−
1525:−
1489:−
1457:−
872:Malliavin
759:Geometric
658:Laplacian
608:Dirichlet
519:Geometric
104:−
51:∫
6569:Calculus
6544:Archived
6369:Archived
5945:function
5878:See also
5225:′
5198:between
5046:′
4636:″
4549:′
4424:velocity
4351:gradient
4231: :
4047:and the
2765:′
2522:′
2174:, where
1958:function
1067:momentum
1059:velocity
997:function
982:calculus
912:Glossary
822:Advanced
800:Jacobian
754:Exterior
684:Gradient
676:Theorems
643:Gradient
582:Integral
544:Binomial
529:Harmonic
394:improper
390:Integral
347:Integral
329:Reynolds
304:Quotient
233:Concepts
69:′
36:Calculus
6410:3614212
5857:(−1, 0)
5849:(−1, 0)
5838:√
5725:circles
5688:equals
5635:
5613:
5601:
5579:
5499:In the
4401:Physics
4361:of the
4113:(or an
4066:(after
3966:0 <
3837:
3819:
3740:. When
2616:, with
1706:, with
1274:is the
1051:physics
1019:to the
1015:of the
907:History
805:Hessian
694:Stokes'
689:Green's
521: (
448: (
392: (
314:Inverse
289:Product
200: (
6508:604533
6506:
6456:
6416:
6408:
6331:, The
6286:
5861:(1, 0)
5853:(1, 0)
5604:, and
5552:. For
5512:, for
5490:, and
4289:) = ±
4188:). If
4117:) and
3970:< 4
3874:Euclid
3725:, and
1124:, and
1081:. The
1075:F = ma
749:Tensor
744:Matrix
631:Vector
549:Taylor
507:Series
144:Limits
6504:JSTOR
6414:S2CID
6406:JSTOR
5922:Notes
5810:) = 0
5779:) = 0
5149:<
5034:with
4970:with
4873:Here
4158:) = 0
4127:or a
4123:is a
4105:is a
3870:Greek
2544:. If
2254:slope
1276:slope
1085:of a
1013:slope
995:of a
572:Ratio
539:Power
458:Euler
436:Discs
431:Parts
299:Power
294:Chain
223:total
6454:ISBN
6335:and
6327:See
6284:ISBN
5859:and
5851:and
5840:1 -
5744:) =
5617:f'''
5455:) +
5440:) +
5403:) +
5331:and
5204:and
5170:and
5138:and
4981:<
4900:and
4783:The
4268:) =
3974:/ 27
3962:/ 27
3794:and
3746:and
3613:are
3593:and
2115:and
2004:and
1071:time
653:Curl
613:Abel
577:Root
6496:doi
6492:110
6398:doi
5818:is
5752:− 1
5642:of
5583:f''
5503:of
4341:In
4299:= 0
4278:= 0
4225:at
4219:of
4209:of
4197:If
4178:at
4166:or
4143:at
4131:of
4109:on
4099:If
4082:to
3958:= 4
3951:+ c
3940:/ 3
3936:= 2
3906:".
3770:at
3758:at
3573:if
3351:As
3302:lim
3225:lim
3122:lim
3041:lim
2962:lim
2896:is
2782:lim
2727:is
2378:lim
2323:As
1813:at
972:In
284:Sum
6560::
6502:.
6490:.
6426:^
6412:.
6404:.
6394:52
6392:.
6351:,
6345:,
6341:,
6307:.
6257:.
6233:.
5836:±
5806:,
5775:,
5760:,
5748:+
5740:,
5694:.
5632:3!
5570:.
5558:,
5541:f'
5484:,
5478:,
5463:−
5448:−
5433:−
5425:+
5411:−
5396:−
5388:+
5374:−
5366:+
5184:))
5176:,
5168:))
5160:,
4655:32
4577:16
4568:32
4490:32
4481:16
4465:16
4419::
4316:f'
4152:f'
4086:.
4051:.
3947:=
3945:ax
3926:–
3924:ax
3850:.
3733:.
3721:,
3691:20
3394:,
2919::
1859::
1537:13
1469:13
1128:.
1120:,
1116:,
1112:,
1046:.
976:,
456:,
452:,
6550:.
6532:.
6530:1
6510:.
6498::
6462:.
6438:.
6420:.
6400::
6317:.
6292:.
6267:.
6243:.
6173:.
6145:x
6142:d
6137:y
6134:d
6101:x
6079:2
6075:y
6071:=
6068:)
6065:y
6062:(
6059:f
6037:2
6033:x
6029:=
6026:)
6023:x
6020:(
6017:f
5997:x
5975:2
5971:x
5967:=
5964:)
5961:x
5958:(
5955:f
5842:x
5831:f
5825:f
5815:f
5808:y
5804:x
5802:(
5800:f
5790:f
5784:f
5777:y
5773:x
5771:(
5769:f
5764:)
5762:y
5758:x
5756:(
5750:y
5746:x
5742:y
5738:x
5736:(
5734:f
5691:f
5685:d
5679:d
5673:f
5663:f
5657:d
5651:d
5645:f
5629:/
5626:)
5624:0
5621:x
5619:(
5607:d
5598:2
5595:/
5592:)
5590:0
5587:x
5585:(
5573:c
5567:f
5561:d
5555:c
5550:)
5548:0
5545:x
5543:(
5535:b
5530:)
5528:0
5525:x
5523:(
5521:f
5515:a
5509:0
5506:x
5493:d
5487:c
5481:b
5475:a
5470:)
5468:0
5465:x
5461:x
5459:(
5457:d
5453:0
5450:x
5446:x
5444:(
5442:c
5438:0
5435:x
5431:x
5429:(
5427:b
5423:a
5418:)
5416:0
5413:x
5409:x
5407:(
5405:c
5401:0
5398:x
5394:x
5392:(
5390:b
5386:a
5381:)
5379:0
5376:x
5372:x
5370:(
5368:b
5364:a
5355:0
5352:x
5347:)
5345:x
5343:(
5341:f
5316:f
5310:f
5303:f
5285:.
5279:a
5273:b
5268:)
5265:a
5262:(
5259:f
5253:)
5250:b
5247:(
5244:f
5238:=
5235:)
5232:c
5229:(
5222:f
5207:b
5201:a
5195:c
5189:f
5182:b
5180:(
5178:f
5174:b
5172:(
5166:a
5164:(
5162:f
5158:a
5156:(
5151:b
5147:a
5141:b
5135:a
5130:)
5128:x
5126:(
5124:f
5117:.
5101:a
5095:b
5090:)
5087:a
5084:(
5081:f
5075:)
5072:b
5069:(
5066:f
5059:=
5056:)
5053:c
5050:(
5043:f
5022:)
5019:b
5016:,
5013:a
5010:(
5004:c
4984:b
4978:a
4957:R
4950:]
4947:b
4944:,
4941:a
4938:[
4935::
4932:f
4903:α
4897:t
4891:x
4886:)
4884:t
4882:,
4880:x
4878:(
4876:u
4858:.
4850:2
4846:x
4837:u
4832:2
4818:=
4812:t
4804:u
4768:.
4760:2
4756:t
4752:d
4747:x
4742:2
4738:d
4731:m
4728:=
4725:)
4722:t
4719:(
4716:F
4658:,
4649:=
4646:)
4643:t
4640:(
4633:x
4629:=
4626:)
4623:t
4620:(
4611:x
4580:,
4574:+
4571:t
4562:=
4559:)
4556:t
4553:(
4546:x
4542:=
4539:)
4536:t
4533:(
4524:x
4493:,
4487:+
4484:t
4478:+
4473:2
4469:t
4459:=
4456:)
4453:t
4450:(
4447:x
4366:"
4297:x
4291:x
4287:x
4285:(
4283:f
4276:x
4270:x
4266:x
4264:(
4262:f
4256:x
4247:x
4238:x
4228:x
4222:f
4212:f
4206:x
4200:f
4191:f
4181:x
4175:f
4156:x
4154:(
4146:x
4140:f
4134:f
4120:x
4111:ℝ
4102:f
3972:a
3968:c
3960:a
3956:c
3949:x
3938:a
3934:x
3928:x
3919:x
3843:f
3833:x
3831:∂
3828:/
3824:y
3822:∂
3809:f
3803:f
3797:y
3791:x
3785:f
3779:f
3773:x
3767:f
3761:x
3755:f
3749:y
3743:x
3699:3
3695:x
3688:=
3683:3
3679:x
3675:)
3672:4
3669:(
3666:5
3663:=
3660:)
3655:4
3651:x
3647:5
3644:(
3638:x
3635:d
3631:d
3601:n
3581:a
3559:1
3553:n
3549:x
3545:n
3542:a
3539:=
3533:x
3530:d
3525:)
3520:n
3516:x
3512:a
3509:(
3506:d
3483:x
3480:2
3477:=
3471:x
3468:d
3463:y
3460:d
3437:x
3434:2
3414:x
3408:+
3405:x
3402:2
3382:0
3362:x
3332:x
3326:+
3323:x
3320:2
3315:0
3309:x
3298:=
3285:x
3275:2
3271:)
3267:x
3261:(
3258:+
3255:x
3249:x
3246:2
3238:0
3232:x
3221:=
3208:x
3198:2
3194:x
3185:2
3181:)
3177:x
3171:(
3168:+
3165:x
3159:x
3156:2
3153:+
3148:2
3144:x
3135:0
3129:x
3118:=
3105:x
3095:2
3091:x
3082:2
3078:)
3074:x
3068:+
3065:x
3062:(
3054:0
3048:x
3037:=
3024:x
3016:)
3013:x
3010:(
3007:f
3001:)
2998:x
2992:+
2989:x
2986:(
2983:f
2975:0
2969:x
2958:=
2948:x
2945:d
2940:y
2937:d
2907:x
2904:2
2882:2
2878:x
2874:=
2871:y
2844:x
2836:)
2833:x
2830:(
2827:f
2821:)
2818:x
2812:+
2809:x
2806:(
2803:f
2795:0
2789:x
2778:=
2775:)
2772:x
2769:(
2762:f
2758:=
2752:x
2749:d
2744:y
2741:d
2715:)
2712:x
2709:(
2706:f
2686:)
2683:x
2680:(
2677:f
2674:=
2671:y
2651:x
2648:d
2624:d
2601:x
2598:d
2593:y
2590:d
2567:)
2564:x
2561:(
2558:f
2555:=
2552:y
2532:)
2529:x
2526:(
2519:f
2498:)
2495:x
2492:(
2489:f
2469:x
2440:x
2432:)
2429:x
2426:(
2423:f
2417:)
2414:x
2408:+
2405:x
2402:(
2399:f
2391:0
2385:x
2354:0
2334:x
2305:x
2297:)
2294:x
2291:(
2288:f
2282:)
2279:x
2273:+
2270:x
2267:(
2264:f
2258:=
2227:x
2219:y
2210:=
2185:x
2162:)
2159:)
2156:x
2150:+
2147:x
2144:(
2141:f
2138:,
2135:x
2129:+
2126:x
2123:(
2103:)
2100:)
2097:x
2094:(
2091:f
2088:,
2085:x
2082:(
2055:2
2051:x
2047:=
2044:y
2024:)
2021:9
2018:,
2015:3
2012:(
1992:)
1989:4
1986:,
1983:2
1980:(
1939:4
1919:)
1916:4
1913:,
1910:2
1907:(
1885:2
1881:x
1877:=
1874:y
1847:4
1827:2
1824:=
1821:x
1801:4
1779:2
1775:x
1771:=
1768:y
1745:2
1741:x
1737:=
1734:y
1691:x
1683:y
1654:x
1644:y
1613:2
1607:=
1601:3
1598:+
1593:6
1584:=
1578:x
1568:y
1534:+
1531:x
1528:2
1522:=
1519:y
1492:2
1466:+
1463:x
1460:2
1454:=
1451:y
1428:x
1418:y
1407:=
1382:x
1362:y
1342:b
1339:+
1336:x
1333:m
1330:=
1327:y
1307:)
1304:)
1301:a
1298:(
1295:f
1292:,
1289:a
1286:(
1262:a
1259:=
1256:x
1236:)
1233:x
1230:(
1227:f
1195:a
1192:=
1189:x
1169:)
1166:x
1163:(
1160:f
1157:=
1154:y
961:e
954:t
947:v
525:)
460:)
396:)
204:)
116:)
113:a
110:(
107:f
101:)
98:b
95:(
92:f
89:=
86:t
83:d
79:)
76:t
73:(
66:f
60:b
55:a
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
