The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.
The "simple" derivative of a function with respect to a variable is denoted either or
often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,
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
is used, with
etc., the corresponding fluxion notation.
When a function depends on more than one variable, a partial derivative
can be used to specify the derivative with respect to one or more variables.
The derivative of a function with respect to the variable is defined as
but may also be calculated more symmetrically as
provided the derivative is known to exist.
It should be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives are most naturally defined in the complex plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist, the same result must be obtained for derivatives taken in any direction in the complex plane. Somewhat surprisingly, almost all of the important functions in mathematics satisfy this property, which is equivalent to saying that they satisfy the Cauchy-Riemann equations.
These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding to the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) of the absolute value function does not exist because at every point in the complex plane, the value of the derivative depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as
As a result of the fact that computer algebra languages and programs such as the Wolfram Language generically deal with complex variables (i.e., the definition of derivative always means complex derivative), correctly returns unevaluated by such software.
If the first derivative exists, the second derivative may be defined as
and calculated more symmetrically as
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 three-dimensional generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "tangent maps."
Performing numerical differentiation is in many ways more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes.
Simple derivatives of some simple functions follow:
where , , etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.
There are a number of important rules for computing derivatives of certain combinations of functions. Derivatives of sums are equal to the sum of derivatives so that
In addition, if is a constant,
The product rule for differentiation states
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,
The quotient rule for derivatives states that
while the power rule gives
Other very important rule for computing derivatives is the chain rule, which states that for ,
or more generally, for
where denotes a partial derivative.
Miscellaneous other derivative identities include
If , where is a constant, then
Derivative identities of inverse functions include
A vector derivative of a vector function
can be defined by
The th derivatives of for , 2, ... are
The th row of the triangle of coefficients 1; 1, 1; 2, 4, 1; 6, 18, 9, 1; ... (OEIS A021009) is given by the absolute values of the coefficients of the Laguerre polynomial .
Faà di Bruno's formula gives an explicit formula for the th derivative of the 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: