# 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[f].
• f'' evaluates to Derivative[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 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:

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:

Introduced in 1988
(1.0)
|
Updated in 1996
(3.0)
1999
(4.0)
2000
(4.1)
2002
(4.2)