# 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

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## Basic Examples(7)

Derivative with respect to x:

 In:= Out= Fourth derivative with respect to x:

 In:= Out= Derivative of order n with respect to x:

 In:= Out= Derivative with respect to x and y:

 In:= Out= Derivative involving a symbolic function f:

 In:= Out= Evaluate derivatives numerically:

 In:= Out= Enter using pd , and subscripts using :

 In:= Out= ## Neat Examples(2)

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