This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

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 times with respect to the first argument, times with respect to the second argument, and so on.
• 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 Mathematica does not know.
• Mathematica attempts to convert Derivative[n][f] and so on to pure functions. Whenever Derivative[n][f] is generated, Mathematica rewrites it as D[f[#], {#, n}]&. If Mathematica finds an explicit value for this derivative, it returns this value. Otherwise, it returns the original Derivative form.
• Derivative[-n][f] represents the indefinite integral of f.
• Derivative[{n1, n2, ...}][f] represents the derivative of taken times with respect to . In general, arguments given in lists in f can be handled by using a corresponding list structure in Derivative.
• N will give a numerical approximation to a derivative.
Derivative of a defined function:
This is equivalent to :
Derivative at a particular value:
This is equivalent to :
The second derivative:
Derivative of a defined function:
 Out[2]=
This is equivalent to :
 Out[3]=
Derivative at a particular value:
 Out[4]=
This is equivalent to :
 Out[5]=
The second derivative:
 Out[6]=
 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: