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In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change.It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.
The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.
Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.
The derivative of at the point is the slope of the tangent to . 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 . The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in by the change in , meaning that . For, the graph of has a slope of , as shown in the diagram below:
For brevity, is often written as , with 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, for instance , 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 slope of the tangent to that point. For example, has a slope of at because the slope of the tangent line to that point is equal to :
The derivative of a function 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 secant line. 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:
The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph and , where is a small number. As before, the slope of the line passing through these two points can be calculated with the formula . This gives
As gets closer and closer to , the slope of the secant line gets closer and closer to the slope of the tangent line. This is formally written as
The above expression means 'as 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 ; this can be written as . If , the derivative can also be written as , with representing an infinitesimal change. For example, represents an infinitesimal change in x. In summary, if , then the derivative of is
provided such a limit exists. is :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 is a proof, using differentiation from first principles, that the derivative of
As approaches , approaches . Therefore, . This proof can be generalised to show that if and are constants. This is known as the power rule. For example, . However, many other functions cannot be differentiated as easily as polynomial functions, meaning that sometimes further techniques are needed to find the derivative of a function. These techniques include the chain rule, product rule, and quotient rule. Other functions cannot be differentiated at all, giving rise to the concept of differentiability.
A closely related concept to the derivative of a function is its differential. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted ∂y/∂x. The linearization of f in all directions at once is called the total derivative.
The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC) and Apollonius of Perga (c. 262–190 BC).Archimedes also made use of indivisibles, although these were primarily used to study areas and volumes rather than derivatives and tangents (see The Method of Mechanical Theorems ).
The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 AD, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the orbit of the Moon.The use of infinitesimals to compute rates of change was developed significantly by Bhāskara 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".
The Islamic mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), in his Treatise on Equations, established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He proved, for example, that the maximum of the cubic ax2 – x3 occurs when x = 2a/3, and concluded therefrom that the equation ax2 — x3 = c has exactly one positive solution when c = 4a3/27, and two positive solutions whenever 0 < c < 4a3/27. The historian of science, Roshdi Rashed, 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.
The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716), who provided independentand unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham (Alhazen). For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Pierre de Fermat (1607-1665), Isaac Barrow (1630–1677), René Descartes (1596–1650), Christiaan Huygens (1629–1695), Blaise Pascal (1623–1662) and John Wallis (1616–1703). Regarding 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. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today.
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 Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.
If f is a differentiable function on ℝ (or an open interval) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero. Points where f'(x) = 0 are called critical points or stationary points (and the value of f at x is called a critical value ). If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.
If f is twice differentiable, then conversely, a critical point x of f can be analysed by considering the second derivative of f at x :
This is called the second derivative test. An alternative approach, called the first derivative test, involves considering the sign of the f' on each side of the critical point.
Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization. By the extreme value theorem, a continuous function on a closed interval 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.
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.
In higher dimensions, a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix 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 "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.
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 shortest path is not immediately clear. These paths are called geodesics, 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 minimal surface and it, too, can be found using the calculus of variations.
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — 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 Newtonian physics:
For example, if an object's position on a line is given by
then the object's velocity is
and the object's acceleration is
which is constant.
A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation
The heat equation in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation
Here u(x,t) is the temperature of the rod at position x and time t and α is a constant that depends on how fast heat diffuses through the rod. (2-3¡)-(3+2)
The mean value theorem gives a relationship between values of the derivative and values of the original function. If f(x) is a real-valued function and a and b are numbers with a < b, then the mean value theorem says that under mild hypotheses, the slope between the two points (a, f(a)) and (b, f(b)) is equal to the slope of the tangent line to f at some point c between a and b. In other words,
In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that f 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 f must equal the slope of one of the tangent lines of f. 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.
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 f(x) at the point x0 is a linear polynomial a + b(x − x0), and it may be possible to get a better approximation by considering a quadratic polynomial a + b(x − x0) + c(x − x0)2. Still better might be a cubic polynomial a + b(x − x0) + c(x − x0)2 + d(x − x0)3, 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 a, b, c, and d that makes the approximation as good as possible.
In the neighbourhood of x0, for a the best possible choice is always f(x0), and for b the best possible choice is always f'(x0). For c, d, and higher-degree coefficients, these coefficients are determined by higher derivatives of f. c should always be f''(x0)/2, and d should always be f'''(x0)/3!. Using these coefficients gives the Taylor polynomial of f. The Taylor polynomial of degree d is the polynomial of degree d which best approximates f, and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f.
The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions. It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there exist smooth functions which are also not analytic.
Some natural geometric shapes, such as circles, cannot be drawn as the graph of a function. For instance, if f(x, y) = x2 + y2 − 1, then the circle is the set of all pairs (x, y) such that f(x, y) = 0. This set is called the zero set of f, and is not the same as the graph of f, which is a paraboloid. The implicit function theorem converts relations such as f(x, y) = 0 into functions. It states that if f is continuously differentiable, then around most points, the zero set of f looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of f. The circle, for instance, can be pasted together from the graphs of the two functions ± √. In a neighborhood of every point on the circle except (−1, 0) and (1, 0), one of these two functions has a graph that looks like the circle. (These two functions also happen to meet (−1, 0) and (1, 0), but this is not guaranteed by the implicit function theorem.)
The implicit function theorem is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together.
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f ) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative is zero. The theorem is named after Michel Rolle.
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as
In mathematics, an implicit equation is a relation of the form R(x1, …, xn) = 0, where R is a function of several variables. For example, the implicit equation of the unit circle is x2 + y2 − 1 = 0.
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.
In numerical analysis, numerical differentiation describes algorithms for estimating the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
Critical point is a wide term used in many branches of mathematics.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:
In mathematics, Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point. Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
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