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 .

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 . 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 , 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^(th) 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 , 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 :

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Fourth derivative with respect to :

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Derivative with respect to and :

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Derivative involving a symbolic function :

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Vector derivative (gradient vector):

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Second-order derivative tensor:

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Evaluate derivatives numerically:

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Enter using EscpdEsc, and subscripts using Control+_:

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