Derivative

f'

represents the derivative of a function f of one argument.

Derivative[n1,n2,…][f]

is the general form, representing a function obtained from f by differentiating n1 times with respect to the first argument, n2 times with respect to the second argument, and so on.

Details

  • f' is equivalent to Derivative[1][f].
  • f'' evaluates to Derivative[2][f].
  • You can think of Derivative as a functional operator which acts on functions to give derivative functions.
  • Derivative is generated when you apply D to functions whose derivatives the Wolfram Language does not know.
  • The Wolfram Language attempts to convert Derivative[n][f] and so on to pure functions. Whenever Derivative[n][f] is generated, the Wolfram Language rewrites it as D[f[#],{#,n}]&. If the Wolfram Language finds an explicit value for this derivative, it returns this value. Otherwise, it returns the original Derivative form.
  • Derivative[-n][f] represents the n ^(th) indefinite integral of f.
  • Derivative[{n1,n2,…}][f] represents the derivative of f[{x1,x2,…}] taken ni times with respect to xi. In general, arguments given in lists in f can be handled by using a corresponding list structure in Derivative.
  • N[f'[x]] will give a numerical approximation to a derivative.

Examples

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Basic Examples  (1)

Derivative of a defined function:

This is equivalent to :

Derivative at a particular value:

This is equivalent to :

The second derivative:

Scope  (5)

The derivative of a function returns a function:

Partial derivatives with respect to different arguments:

The partial derivative with respect to the first argument:

A mixed partial evaluated at a particular value:

Partial derivatives for functions with list arguments:

The partial derivative with respect to the first element:

A mixed partial evaluated at a particular value:

Define a derivative for a function:

Define partial derivatives for a function:

This effectively defines the gradient:

Show the vector field:

Generalizations & Extensions  (1)

Derivative with a negative integer order can do integrals:

Properties & Relations  (1)

Use N to find a numerical approximation to the derivative:

Wolfram Research (1988), Derivative, Wolfram Language function, https://reference.wolfram.com/language/ref/Derivative.html (updated 2002).

Text

Wolfram Research (1988), Derivative, Wolfram Language function, https://reference.wolfram.com/language/ref/Derivative.html (updated 2002).

CMS

Wolfram Language. 1988. "Derivative." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/Derivative.html.

APA

Wolfram Language. (1988). Derivative. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Derivative.html

BibTeX

@misc{reference.wolfram_2024_derivative, author="Wolfram Research", title="{Derivative}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Derivative.html}", note=[Accessed: 18-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_derivative, organization={Wolfram Research}, title={Derivative}, year={2002}, url={https://reference.wolfram.com/language/ref/Derivative.html}, note=[Accessed: 18-December-2024 ]}