# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# D

D[f,x]
gives the partial derivative .

D[f,{x,n}]
gives the multiple derivative .

D[f,x,y,]
differentiates f successively with respect to x,y,.

D[f,{{x1,x2,}}]
for a scalar f gives the vector derivative .

D[f,{array}]
gives a tensor derivative.

## Details and OptionsDetails and Options

• D[f,x] can be input as . The character is entered as EscpdEsc or [PartialD]. The variable x is entered as a subscript.
• All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
• D[f,var1,,NonConstants->{u1,}] specifies that every implicitly depends on every varj, so that they do not have zero partial derivative.
• D[f,] threads over lists that appear in f.
• D[f,{list}] effectively threads D over each element of list.
• D[f,{list,n}] is equivalent to D[f,{list},{list},] where {list} is repeated n times. If f is a scalar, and list has depth 1, then the result is a tensor of rank n, as in the n term of the multivariate Taylor series of f.
• D[f,{list1},{list2},] is normally equivalent to First[Outer[D,{f},list1,list2,]].
• If f is a list, then D[f,{list}] effectively threads first over each element of f, and then over each element of list. The result is an array with dimensions Join[Dimensions[f],Dimensions[list]].
• Numerical approximations to derivatives can be found using N.
• D uses the chain rule to simplify derivatives of unknown functions.
• D[f,x,y] can be input as . The character [InvisibleComma], entered as Esc,Esc, can be used instead of an ordinary comma. It does not display, but is still interpreted just like a comma.
• If any of the arguments to D are SparseArray objects, the result will be a SparseArray object. »

## ExamplesExamplesopen allclose all

### Basic Examples  (7)Basic Examples  (7)

Derivative with respect to x:

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Fourth derivative with respect to x:

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Derivative with respect to x and y:

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Derivative involving a symbolic function f:

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Secondorder derivative tensor:

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Evaluate derivatives numerically:

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Enter using EscpdEsc, and subscripts using Control+_:

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