D
Usage Message: D[f, x] gives the partial derivative of f with respect to x. D[f, {x, n}] gives the nth partial
derivative of f with respect to x. D[f, x1, x2, ... ] gives a mixed derivative. Attributes[D] = {Protected,
ReadProtected} Options: NonConstants
-> {} Related Symbols: Derivative
ND
Notes: If an expression contains functions that are not symbolically differentiable, the result will be expressed using Derivative
. Expressions involving Derivative
are normally displayed in special notations. The Derivative
structure can be seen using FullForm
or InputForm
:
In[1]:= expr = D[f[x], x]
Out[1]= f'[x]
In[2]:= InputForm[expr]
Out[2]//InputForm= Derivative[1][f][x]
Derivatives of expressions that are not differentiable as functions of a complex variable will be represented in terms of Derivative
. For a function to be differentiable, the derivative must be the same in all directions in the complex plane. For example, Abs
is not a differentiable function.
In[3]:= D[Abs[x], x]
Out[3]= Abs'[x]
The non-differentiability of Abs
can be seen by observing (using numerical approximations of the derivatives) that the derivative is different in different directions.
In[4]:= Block[{epsilon = 0.000001},
(Abs[1.7 + epsilon] - Abs[1.7])/epsilon]
Out[4]= 1.
In[5]:= Block[{epsilon = 0.000001 (1 + I)},
(Abs[1.7 + epsilon] - Abs[1.7])/epsilon]
Out[5]= 0.5 - 0.5 I
In[6]:= Block[{epsilon = 0.000001 (-1 + I)},
(Abs[1.7 + epsilon] - Abs[1.7])/epsilon]
Out[6]= 0.5 + 0.5 I
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/D.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/D.html
Questions or comments? Send email to support@wolfram.com.
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