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D
D[
f
,
x
] gives the partial derivative  .
D[
f
,
x
,
n
] gives the multiple derivative  .
D[
f
,
,
, ... ] gives  .
D[
f
,
x
] can be input as  . The character  is entered as  pd or \[PartialD]. The variable x is entered as a subscript.
All quantities that do not explicitly depend on the  are taken to have zero partial derivative.
D[
f
,
, ... ,
NonConstants
->
, ... 
] specifies that the  implicitly depend on the  , so that they do not have zero partial derivative.
The derivatives of built-in mathematical functions are evaluated when possible in terms of other built-in mathematical functions.
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 ,, can be used instead of an ordinary comma. It does not display, but is still interpreted just like a comma.
See the Mathematica book: Section 1.5.2, Section 3.5.1.
See also: Dt, Derivative.
Related package: Calculus`VectorAnalysis`, NumericalMath`NLimit`.
Further Examples
Here is the derivative of  with respect to x.
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Here is the familiar Chain Rule of first year calculus.
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This gives the fourth derivative of  .
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Here is the partial derivative  .
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Normally, if you differentiate a function with respect to x, say, Mathematica will treat all other parameters as constants.
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By specifying that t depends upon x, you can get the desired result for such expressions.
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Here are some advanced examples.
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Sometimes the derivative is kept unevaluated until an argument is substituted for which an evaluation can be given.
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