This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# D

 Dgives the partial derivative . Dgives the multiple derivative . Ddifferentiates f successively with respect to . Dfor a scalar f gives the vector derivative . Dgives a tensor derivative.
• D can be input as . The character is entered as Esc pd Esc 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 threads over lists that appear in f.
• D effectively threads D over each element of list.
• D is equivalent to D 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 is normally equivalent to First[Outer[D, {f}, list1, list2, ...]].
• If f is a list, then D 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 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.
Derivative with respect to :
Fourth derivative with respect to :
Derivative with respect to and :
Derivative involving a symbolic function :
Second-order derivative tensor:
Evaluate derivatives numerically:
Enter using Esc pd Esc, and subscripts using Control+_:
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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Second-order derivative tensor:
 Out[2]=

Evaluate derivatives numerically:
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Enter using Esc pd Esc, and subscripts using Control+_:
 Out[1]=
 Scope   (2)
Differentiate with respect to different formal variables:
 Options   (1)
Differentiate with y considered as depending on x:
Solve for the derivative of y to effect implicit differentiation:
 Applications   (5)
Find the turning points on a plane curve:
Perform the change of variable in an integral:
Find the curvature of a circular helix with radius r and pitch c:
Compute the coefficients of a power series:
Construct the differential equation satisfied by an implicit function :
Results may not immediately be given in the simplest possible form:
Functions given in different forms can yield the same derivatives: