{"history":[{"id":"2f0e89fe-4ce5-4fb5-846b-eb9d5a4f09f1","wordChanged":"cannot","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **differential [calculus](https://bliptext.com/articles/calculus)** is a subfield of [calculus](https://bliptext.com/articles/calculus) that studies the rates at which quantities change.\nIt is one of the two traditional divisions of calculus, the other being [integral calculus](https://bliptext.com/articles/integral-calculus)—the study of the area above a curve.\nThe primary objects of study in [differential](https://bliptext.com/articles/differential-of-a-function) calculus are the [derivative](https://bliptext.com/articles/derivative) of a [function](https://bliptext.com/articles/function-mathematics), related notions such as the [differential](https://bliptext.com/articles/differential-of-a-function), and their applications.\nThe derivative of a function at a chosen input value describes the [rate of change](https://bliptext.com/articles/rate-mathematics) of the function near that input value.\nThe process of finding a derivative is called **differentiation**.\nGeometrically, the derivative at a point is the [slope](https://bliptext.com/articles/slope) of the [tangent line](https://bliptext.com/articles/tangent) to the [graph of the function](https://bliptext.com/articles/graph-of-a-function) at that point, provided that the derivative exists and is defined at that point.\nFor a [real-valued function](https://bliptext.com/articles/real-valued-function) of a single real variable, the derivative of a function at a point generally determines the best [linear approximation](https://bliptext.com/articles/linear-approximation) to the function at that point.\nDifferential calculus and [integral calculus](https://bliptext.com/articles/integral-calculus) are connected by the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus).\nThis states that differentiation is the reverse process to [integration](https://bliptext.com/articles/integration-mathematics).\nDifferentiation has applications in nearly all quantitative disciplines.\nIn [physics](https://bliptext.com/articles/physics), the derivative of the [displacement](https://bliptext.com/articles/displacement-vector) of a moving body with respect to time is the [velocity](https://bliptext.com/articles/velocity) of the body, and the derivative of the [velocity](https://bliptext.com/articles/velocity) with respect to time is [acceleration](https://bliptext.com/articles/acceleration).\nThe derivative of the [momentum](https://bliptext.com/articles/momentum) of a body with respect to [time](https://bliptext.com/articles/time) equals the force applied to the body; rearranging this derivative statement leads to the famous\nequation associated with [Newton's second law of motion](https://bliptext.com/articles/newton-s-laws-of-motion).\nThe [reaction rate](https://bliptext.com/articles/reaction-rate) of a [chemical reaction](https://bliptext.com/articles/chemical-reaction) is a derivative.\nIn [operations research](https://bliptext.com/articles/operations-research), derivatives determine the most efficient ways to transport materials and design factories.\nDerivatives are frequently used to find the [maxima and minima](https://bliptext.com/articles/maxima-and-minima) of a function.\nEquations involving derivatives are called [differential equations](https://bliptext.com/articles/differential-equations) and are fundamental in describing [natural phenomena](https://bliptext.com/articles/natural-phenomenon).\nDerivatives and their generalizations appear in many fields of mathematics, such as [complex analysis](https://bliptext.com/articles/complex-analysis), [functional analysis](https://bliptext.com/articles/functional-analysis), [differential geometry](https://bliptext.com/articles/differential-geometry), [measure theory](https://bliptext.com/articles/measure-theory), and [abstract algebra](https://bliptext.com/articles/abstract-algebra).\n# [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. | upload.wikimedia.org/wikipedia/commons/0/0f/Tangent///to///a///curve.svg]\n\n# Derivative\nThe derivative of f(x) at the point x=a is the [slope](https://bliptext.com/articles/slope) of the tangent to (a,f(a)).\nIn order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form y=mx+b.\nThe slope of an equation is its steepness.\nIt can be found by picking any two points and dividing the change in y by the change in x, meaning that.\nFor, the graph of y=-2x+13 has a slope of -2, as shown in the diagram below:\nFor brevity, is often written as, with \\Delta being the Greek letter delta, meaning 'change in'.\nThe slope of a linear equation is constant, meaning that the steepness is the same everywhere.\nHowever, many graphs such as y=x^2 vary in their steepness.\nThis means that you can no longer pick any two arbitrary points and compute the slope.\nInstead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point.\nThe slope of a curve at a particular point is equal to the slope of the tangent to that point.\nFor example, y=x^2 has a slope of 4 at x=2 because the slope of the tangent line to that point is equal to 4:\nThe derivative of a [function](https://bliptext.com/articles/function-mathematics) is then simply the slope of this tangent line.\nEven 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.\nThis is known as a [secant line](https://bliptext.com/articles/secant-line).\nIf 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:\nThe advantage of using a secant line is that its slope can be calculated directly.\nConsider the two points on the graph (x,f(x)) and, where \\Delta x is a small number.\nAs before, the slope of the line passing through these two points can be calculated with the formula.\nThis gives\nAs \\Delta x gets closer and closer to 0, the slope of the secant line gets closer and closer to the slope of the tangent line.\nThis is formally written as\nThe above expression means 'as \\Delta x gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'.\nThe value that is being approached is the derivative of f(x); this can be written as f'(x).\nIf y=f(x), the derivative can also be written as, with d representing an [infinitesimal](https://bliptext.com/articles/infinitesimal) change.\nFor example, dx represents an infinitesimal change in x. In summary, if y=f(x), then the derivative of f(x) is\nprovided such a limit exists.\nWe 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.\nDifferentiating a function using the above definition is known as differentiation from first principles.\nHere is a proof, using differentiation from first principles, that the derivative of y=x^2 is 2x:\nAs \\Delta x approaches 0, 2x+\\Delta x approaches 2x.\nTherefore,.\nThis proof can be generalised to show that if a and n are [constants](https://bliptext.com/articles/constant-mathematics).\nThis is known as the [power rule](https://bliptext.com/articles/power-rule).\nFor example,.\nHowever, many other functions cannot be differentiated as easily as [polynomial functions](https://bliptext.com/articles/polynomial), meaning that sometimes further techniques are needed to find the derivative of a function.\nThese techniques include the [chain rule](https://bliptext.com/articles/chain-rule), [product rule](https://bliptext.com/articles/product-rule), and [quotient rule](https://bliptext.com/articles/quotient-rule).\nOther functions cannot be differentiated at all, giving rise to the concept of [differentiability](https://bliptext.com/articles/differentiability).\nA closely related concept to the derivative of a function is its [differential](https://bliptext.com/articles/differential-mathematics).\nWhen\nx\nand\ny\nare real variables, the derivative of\nf\nat\nx\nis the slope of the tangent line to the graph of\nf\nat\nx\n. Because the source and target of\nf\nare one-dimensional, the derivative of\nf\nis a real number.\nIf\nx\nand\ny\nare vectors, then the best linear approximation to the graph of\nf\ndepends on how\nf\nchanges in several directions at once.\nTaking the best linear approximation in a single direction determines a [partial derivative](https://bliptext.com/articles/partial-derivative), which is usually denoted\n∂y⁄∂x\n. The linearization of\nf\nin all directions at once is called the [total derivative](https://bliptext.com/articles/total-derivative).\n# [differential calculus | upload.wikimedia.org/wikipedia/commons/6/65/Curve///with///tangent///line.png]\n# [differential calculus | upload.wikimedia.org/wikipedia/commons/9/9f/The///graph///of///y%3D-2x%2B13.png]\n# [differential calculus | upload.wikimedia.org/wikipedia/commons/c/c3/Graph///of///the///function///f%28x%29%3Dx%5E2///with///a///tangent///line///drawn///to///%282%2C4%29.png]\n# [The dotted line goes through the points (2,4) and (3,9), which both lie on the curve y=x^2. 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. | upload.wikimedia.org/wikipedia/commons/8/8e/Tangent///line///versus///secant///line.png]\n\n# History of differentiation\nThe concept of a derivative in the sense of a [tangent line](https://bliptext.com/articles/tangent-line) is a very old one, familiar to ancient [Greek](https://bliptext.com/articles/ancient-greece) mathematicians such as [Euclid](https://bliptext.com/articles/euclid) (c. 300 BC), [Archimedes](https://bliptext.com/articles/archimedes) (c. 287–212 BC), and [Apollonius of Perga](https://bliptext.com/articles/apollonius-of-perga) (c. 262–190 BC).\n[Archimedes](https://bliptext.com/articles/archimedes) also made use of [indivisibles](https://bliptext.com/articles/cavalieri-s-principle), although these were primarily used to study areas and volumes rather than derivatives and tangents (see [The Method of Mechanical Theorems](https://bliptext.com/articles/the-method-of-mechanical-theorems)).\nThe use of infinitesimals to compute rates of change was developed significantly by [Bhāskara II](https://bliptext.com/articles/bh-skara-ii) (1114–1185); indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as \"[Rolle's theorem](https://bliptext.com/articles/rolle-s-theorem)\".\nThe mathematician, [Sharaf al-Dīn al-Tūsī](https://bliptext.com/articles/sharaf-al-d-n-al-t-s) (1135–1213), in his Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials.\nHe obtained, for example, that the maximum (for positive x) of the cubic\nax2 – x3\noccurs when\n, and concluded therefrom that the equation\nhas exactly one positive solution when\n, and two positive solutions whenever\n0 < c < 4a3 / 27\n. The historian of science, [Roshdi Rashed](https://bliptext.com/articles/roshdi-rashed), has argued that al-Tūsī must have used the derivative of the cubic to obtain this result.\nRashed'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.\nThe modern development of calculus is usually credited to [Isaac Newton](https://bliptext.com/articles/isaac-newton) (1643–1727) and [Gottfried Wilhelm Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz) (1646–1716), who provided independent and unified approaches to differentiation and derivatives.\nThe key insight, however, that earned them this credit, was the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus) relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes.\nFor their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as [Pierre de Fermat](https://bliptext.com/articles/pierre-de-fermat) (1607-1665), [Isaac Barrow](https://bliptext.com/articles/isaac-barrow) (1630–1677), [René Descartes](https://bliptext.com/articles/ren-descartes) (1596–1650), [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens) (1629–1695), [Blaise Pascal](https://bliptext.com/articles/blaise-pascal) (1623–1662) and [John Wallis](https://bliptext.com/articles/john-wallis) (1616–1703).\nRegarding Fermat's influence, Newton once wrote in a letter that \"I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general.\" Isaac Barrow is generally given credit for the early development of the derivative.\nNevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to [theoretical physics](https://bliptext.com/articles/theoretical-physics), while Leibniz systematically developed much of the notation still used today.\nSince the 17th century many mathematicians have contributed to the theory of differentiation.\nIn the 19th century, calculus was put on a much more rigorous footing by mathematicians such as [Augustin Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) (1789–1857), [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) (1826–1866), and [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) (1815–1897).\nIt was also during this period that the differentiation was generalized to [Euclidean space](https://bliptext.com/articles/euclidean-space) and the [complex plane](https://bliptext.com/articles/complex-plane).\nThe 20th century brought two major steps towards our present understanding and practice of derivation : [Lebesgue integration](https://bliptext.com/articles/lebesgue-integration), besides extending integral calculus to many more functions, clarified the relation between derivation and integration with the notion of [absolute continuity](https://bliptext.com/articles/absolute-continuity).\nLater the [theory of distributions](https://bliptext.com/articles/theory-of-distributions) (after [Laurent Schwartz](https://bliptext.com/articles/laurent-schwartz)) extended derivation to generalized functions (e.g., the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function) previously introduced in [Quantum Mechanics](https://bliptext.com/articles/quantum-mechanics)) and became fundamental to nowadays applied analysis especially by the use of [weak solutions](https://bliptext.com/articles/weak-solution) to [partial differential equations](https://bliptext.com/articles/partial-differential-equations).\n\n# Applications of derivatives\n\n## Optimization\nIf\nf\nis a [differentiable function](https://bliptext.com/articles/differentiable-function) on\nℝ\n(or an [open interval](https://bliptext.com/articles/open-interval)) and\nx\nis a [local maximum](https://bliptext.com/articles/local-maximum) or a [local minimum](https://bliptext.com/articles/local-minimum) of\nf\n, then the derivative of\nf\nat\nx\nis zero.\nPoints where\nare called [critical points](https://bliptext.com/articles/critical-point-mathematics) or [stationary points](https://bliptext.com/articles/stationary-point) (and the value of\nf\nat\nx\nis called a **critical value**). If\nf\nis not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.\nIf\nf\nis twice differentiable, then conversely, a critical point\nx\nof\nf\ncan be analysed by considering the [second derivative](https://bliptext.com/articles/second-derivative) of\nf\nat\nx\nx\nis a local minimum;\nx\nis a local maximum;\nx\ncould be a local minimum, a local maximum, or neither.\n(For example,\nhas a critical point at\n, but it has neither a maximum nor a minimum there, whereas\nhas a critical point at\nand a minimum and a maximum, respectively, there.) This is called the [second derivative test](https://bliptext.com/articles/second-derivative-test). An alternative approach, called the [first derivative test](https://bliptext.com/articles/first-derivative-test), involves considering the sign of the\nf '\non each side of the critical point.\nTaking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in [optimization](https://bliptext.com/articles/optimization-mathematics).\nBy the [extreme value theorem](https://bliptext.com/articles/extreme-value-theorem), a continuous function on a [closed interval](https://bliptext.com/articles/closed-interval) must attain its minimum and maximum values at least once.\nIf the function is differentiable, the minima and maxima can only occur at critical points or endpoints.\nThis 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.\nIn [higher dimensions](https://bliptext.com/articles/higher-dimension), a critical point of a [scalar valued](https://bliptext.com/articles/scalar-mathematics) function is a point at which the [gradient](https://bliptext.com/articles/gradient) is zero.\nThe [second derivative](https://bliptext.com/articles/second-derivative) test can still be used to analyse critical points by considering the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the [Hessian matrix](https://bliptext.com/articles/hessian-matrix) of second partial derivatives of the function at the critical point.\nIf all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.\nIf there are some positive and some negative eigenvalues, then the critical point is called a \"[saddle point](https://bliptext.com/articles/saddle-point)\", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.\n\n### Calculus of optimizations\nOne 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.\nIf the surface is a plane, then the shortest curve is a line.\nBut if the surface is, for example, egg-shaped, then the [shortest path](https://bliptext.com/articles/shortest-path-problem) is not immediately clear.\nThese paths are called [geodesics](https://bliptext.com/articles/geodesic), and one of the most fundamental problems in the calculus of variations is finding [geodesics](https://bliptext.com/articles/geodesic).\nAnother example is: Find the smallest area surface filling in a closed curve in space.\nThis surface is called a [minimal surface](https://bliptext.com/articles/minimal-surface) and it, too, can be found using the calculus of variations.\n\n## Physics\nCalculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called [differential equations](https://bliptext.com/articles/differential-equation).\nPhysics is particularly concerned with the way quantities change and develop over time, and the concept of the \"**[time derivative](https://bliptext.com/articles/time-derivative)**\" — the rate of change over time — is essential for the precise definition of several important concepts.\nIn particular, the time derivatives of an object's position are significant in [Newtonian physics](https://bliptext.com/articles/newtonian-physics):\nFor example, if an object's position on a line is given by\nthen the object's velocity is\nand the object's acceleration is\nwhich is constant.\n\n## Differential equations\nA differential equation is a relation between a collection of functions and their derivatives.\nAn [ordinary differential equation](https://bliptext.com/articles/ordinary-differential-equation) is a differential equation that relates functions of one variable to their derivatives with respect to that variable.\nA [partial differential equation](https://bliptext.com/articles/partial-differential-equation) is a differential equation that relates functions of more than one variable to their [partial derivatives](https://bliptext.com/articles/partial-derivative).\nDifferential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself.\nFor example, [Newton's second law](https://bliptext.com/articles/newton-s-second-law), which describes the relationship between acceleration and force, can be stated as the ordinary differential equation\nThe [heat equation](https://bliptext.com/articles/heat-equation) in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation\nHere\nu(x,t)\nis the temperature of the rod at position\nx\nand time\nt\nand\nα\nis a constant that depends on how fast heat diffuses through the rod.\n\n## Mean value theorem\nThe mean value theorem gives a relationship between values of the derivative and values of the original function.\nIf\nf(x)\nis a real-valued function and\na\nand\nb\nare numbers with\na < b\n, then the mean value theorem says that under mild hypotheses, the slope between the two points\n(a, f(a))\nand\n(b, f(b))\nis equal to the slope of the tangent line to\nf\nat some point\nc\nbetween\na\nand\nb\n. In other words,\nIn practice, what the mean value theorem does is control a function in terms of its derivative.\nFor instance, suppose that\nf\nhas derivative equal to zero at each point.\nThis means that its tangent line is horizontal at every point, so the function should also be horizontal.\nThe mean value theorem proves that this must be true: The slope between any two points on the graph of\nf\nmust equal the slope of one of the tangent lines of\nf\n. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero.\nBut that says that the function does not move up or down, so it must be a horizontal line.\nMore complicated conditions on the derivative lead to less precise but still highly useful information about the original function.\n# [The mean value theorem: For each differentiable function with a