D

D[f,x]

gives the partial derivative .

D[f,{x,n}]

gives the multiple derivative .

D[f,x,y,]

gives the partial derivative .

D[f,{x,n},{y,m},]

gives the multiple partial derivative .

D[f,{{x1,x2,}}]

for a scalar f gives the vector derivative .

D[f,{array}]

gives an array derivative.

Details and Options

  • D is also known as derivative for univariate functions.
  • By using the character , entered as pd or \[PartialD], with subscripts, derivatives can be entered as follows:
  • D[f,x]xf
    D[f,{x,n}]{x,n}f
    D[f,x,y]x,yf
    D[f,{{x,y}}]{{x,y}}f
  • The comma can be made invisible by using the character \[InvisibleComma] or ,.
  • The partial derivative D[f[x],x] is defined as , and higher derivatives D[f[x,y],x,y] are defined recursively as etc.
  • The order of derivatives n and m can be symbolic and they are assumed to be positive integers.
  • The derivative D[f[x],{x,n}] for a symbolic f is represented as Derivative[n][f][x].
  • For some functions f, Derivative[n][f][x] may not be known, but can be approximated by applying N. »
  • New derivative rules can be added by adding values to Derivative[n][f][x]. »
  • For lists, D[{f1,f2,},x] is equivalent to {D[f1,x],D[f2,x],} recursively. »
  • D[f,{array}] effectively threads D over each element of array.
  • D[f,{array,n}] is equivalent to D[f,{array},{array},], where {array} is repeated n times.
  • D[f,{array1},{array2},] is normally equivalent to First[Outer[D,{f},array1,array2,]]. »
  • Common array derivatives include:
  • D[f,{{x1,x2,}}]gradient{D[f,x1],D[f,x2],}
    D[f,{{x1,x2,},2}]Hessian{{D[f,x1,x1],D[f,x1,x2],},{D[f,x2,x1],D[f,x2,x2],},}
    D[{f1,f2,},{{x1,x2,}}]Jacobian{{D[f1,x1],D[f1,x2],},
    {D[f2,x1],D[f2,x2],},}
  • If f is a scalar and x={x1,}, then the multivariate Taylor series at x0={x01,} is given by:
  • ,
  • where fi=D[f,{x,i}]/.{x1x01,} is an array with tensor rank . »
  • If f and x are both arrays, then D[f,{x}] effectively threads first over each element of f, and then over each element of x. The result is an array with dimensions Join[Dimensions[f],Dimensions[x]]. »
  • D can formally differentiate operators such as integrals and sums, taking into account scoped variables as well as the structure of the particular operator.
  • Examples of operator derivatives include:
  • is not scoped by the integral
    is scoped by the integral
    is not scoped by the integral transform
    is scoped by by the integral transform
  • All expressions that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • The setting NonConstants{u1,} specifies that ui depends on all variables x, y, etc. and does not have zero partial derivative. »

Examples

open all close all

Basic Examples  (7)

Derivative with respect to x:

In[1]:=
Click for copyable input
Out[1]=

Fourth derivative with respect to x:

In[1]:=
Click for copyable input
Out[1]=

Derivative of order n with respect to x:

In[1]:=
Click for copyable input
Out[1]=

Derivative with respect to x and y:

In[1]:=
Click for copyable input
Out[1]=

Derivative involving a symbolic function f:

In[1]:=
Click for copyable input
Out[1]=

Evaluate derivatives numerically:

In[1]:=
Click for copyable input
Out[1]=

Enter using pd, and subscripts using :

In[1]:=
Click for copyable input
Out[1]=

Scope  (82)

Options  (1)

Applications  (41)

Properties & Relations  (21)

Possible Issues  (4)

Interactive Examples  (2)

Neat Examples  (2)

Introduced in 1988
(1.0)
|
Updated in 2017
(11.1)