Derivative

The derivative of a function is its instantaneous rate of change with respect to one of its variables.

The "simple" derivative of a function with respect to a variable is denoted either or

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often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,

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The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).

When a derivative is taken times, the notation or

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is used, with

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etc., the corresponding fluxion notation.

When a function depends on more than one variable, a partial derivative

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can be used to specify the derivative with respect to one or more variables.

The derivative of a real-valued function on a real interval, at an interior point , is defined by the finite limit

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but may also be calculated more symmetrically as

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provided the derivative is known to exist.

The preceding definitions describe real derivatives, for which approaches 0 through real values along the real axis. Elementary calculus texts commonly consider functions of a real variable, so this restriction is often left implicit.

For a function of a complex variable, a complex derivative requires that the difference quotient approach the same finite limit as along every path in the complex plane. Agreement along straight lines alone is insufficient. The Cauchy-Riemann equations are necessary; they are sufficient if the real and imaginary components are differentiable as functions of two real variables at the point.

These considerations can lead to confusion for students when they first encounter complex derivatives. For example, the familiar derivative of the real absolute value function does not imply that has a complex derivative. The absolute value function has no complex derivative at any point in the complex plane. However, the real absolute value function has derivative

(8)

The two nonzero cases are the slopes of the two straight pieces of its function graph. At , the left and right slopes disagree, so the real derivative does not exist there. Thus the real derivative is valid away from the corner even though the corresponding complex derivative does not exist.

In the Wolfram Language, RealAbs[x] supports differentiation of the absolute value function with respect to a real variable.

If the first derivative exists, the second derivative may be defined as

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and calculated more symmetrically as

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again provided the second derivative is known to exist.

Note that in order for the limit to exist, both and must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

A generalization of the derivative to an arbitrary direction is the directional derivative. The derivative of a smooth function between manifolds is represented by a tangent map between the corresponding tangent spaces.

Numerical differentiation can amplify small errors in function values, since a finite difference estimate divides a difference of values by a small step size.

Real-variable derivatives of some functions follow, at points where they are differentiable. For the power rule with real exponent , take ; for , take . The inverse sine and inverse cosine formulas require , the inverse secant and inverse cosecant formulas require , and the inverse cotangent formula uses the MathWorld branch and excludes :

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where , , etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.

For differentiable functions, derivatives of sums are equal to the sum of derivatives so that

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In addition, if is a constant,

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The product rule for differentiation states

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where denotes the derivative of with respect to . This derivative rule can be applied iteratively to yield derivative rules for products of three or more functions, for example,

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The quotient rule for derivatives states that, for ,

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while the power rule gives

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Another rule for computing derivatives is the chain rule, which states that for ,

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or more generally, for a differentiable function and differentiable functions and ,

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where denotes a partial derivative.

Miscellaneous other derivative identities, requiring the displayed denominators to be nonzero, include

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If , where is a constant, is continuously differentiable, and , then

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so

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Derivative identities of inverse functions, for a three-times continuously differentiable function with , include

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A vector derivative of a vector function

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can be defined by

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The th derivatives of for , 2, ... are

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The triangle of coefficients 1; 1, 1; 2, 4, 1; 6, 18, 9, 1; ... gives the absolute values of the coefficients of , with rows indexed from , where is a Laguerre polynomial. The signed triangle is given by OEIS A021009.

Faà di Bruno's formula gives an explicit formula for the th derivative of the function composition .

The June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007) featured the following derivative as a "hard" exam problem intended for a remedial math class but accidentally handed out to the normal class. For and real ,

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