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Derivative
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Derivative

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f'

represents the derivative of a function f of one argument.

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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)Summary of the most common use cases

Derivative of a defined function:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-l90fis
In[2]:=2
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https://wolfram.com/xid/0ftvje6-b0zx9b
Out[2]=2

This is equivalent to :

In[3]:=3
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https://wolfram.com/xid/0ftvje6-eu8c3
Out[3]=3

Derivative at a particular value:

In[4]:=4
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https://wolfram.com/xid/0ftvje6-ghwu9f
Out[4]=4

This is equivalent to :

In[5]:=5
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https://wolfram.com/xid/0ftvje6-kgtvlq
Out[5]=5

The second derivative:

In[6]:=6
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https://wolfram.com/xid/0ftvje6-3y89x
Out[6]=6

Scope  (5)Survey of the scope of standard use cases

The derivative of a function returns a function:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-ifzu1p
In[2]:=2
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https://wolfram.com/xid/0ftvje6-dh95rk
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ftvje6-ep2n9u
Out[3]=3

Partial derivatives with respect to different arguments:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-cn1wg

The partial derivative with respect to the first argument:

In[2]:=2
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https://wolfram.com/xid/0ftvje6-cnag1a
Out[2]=2

A mixed partial evaluated at a particular value:

In[3]:=3
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https://wolfram.com/xid/0ftvje6-pjvok8
Out[3]=3

Partial derivatives for functions with list arguments:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-c9uwoq

The partial derivative with respect to the first element:

In[2]:=2
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https://wolfram.com/xid/0ftvje6-go5yo7
Out[2]=2

A mixed partial evaluated at a particular value:

In[3]:=3
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https://wolfram.com/xid/0ftvje6-d36ifb
Out[3]=3

Define a derivative for a function:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-83do7
In[2]:=2
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https://wolfram.com/xid/0ftvje6-e3siw2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ftvje6-hk2nzb
Out[3]=3

Define partial derivatives for a function:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-fzlnu5

This effectively defines the gradient:

In[2]:=2
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https://wolfram.com/xid/0ftvje6-i89hz9
Out[2]=2

Show the vector field:

In[3]:=3
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https://wolfram.com/xid/0ftvje6-g7mqjl
Out[3]=3

Generalizations & Extensions  (1)Generalized and extended use cases

Derivative with a negative integer order can do integrals:

In[1]:=1
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https://wolfram.com/xid/0ftvje6-llu
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvje6-txb
Out[2]=2

Properties & Relations  (1)Properties of the function, and connections to other functions

Use N to find a numerical approximation to the derivative:

In[1]:=1
✖
https://wolfram.com/xid/0ftvje6-0h1yq
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvje6-g7igxm
Out[2]=2
Wolfram Research (1988), Derivative, Wolfram Language function, https://reference.wolfram.com/language/ref/Derivative.html (updated 2002).
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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).

✖
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.

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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

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

BibTeX

@misc{reference.wolfram_2025_derivative, author="Wolfram Research", title="{Derivative}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Derivative.html}", note=[Accessed: 09-April-2025 ]}

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@misc{reference.wolfram_2025_derivative, author="Wolfram Research", title="{Derivative}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/Derivative.html}", note=[Accessed: 09-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_derivative, organization={Wolfram Research}, title={Derivative}, year={2002}, url={https://reference.wolfram.com/language/ref/Derivative.html}, note=[Accessed: 09-April-2025 ]}

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@online{reference.wolfram_2025_derivative, organization={Wolfram Research}, title={Derivative}, year={2002}, url={https://reference.wolfram.com/language/ref/Derivative.html}, note=[Accessed: 09-April-2025 ]}