Differentiation
| D[f,x] | partial derivative  |
| D[f,x,y,...] | multiple derivative  |
| D[f{x,n}] | n derivative  |
| D[f,x,NonConstants->{v1,v2,...}] |  with the  taken to depend on x |
Partial differentiation operations.
This gives
.
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This gives the third derivative.
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You can differentiate with respect to any expression that does not involve explicit mathematical operations.
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D does
partial differentiation. It assumes here that
is independent of
.
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Instead of giving an explicit function
, you can tell
D that
implicitly depends on
.
D[y, x, NonConstants->{y}] then represents
, with
implicitly depending on
.
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| D[f,{{x1,x2,...}}] | the gradient of a scalar function  |
| D[f,{{x1,x2,...},2}] | the Hessian matrix for f |
| D[f,{{x1,x2,...},n}] | the n -order Taylor series coefficient |
| D[{f1,f2,...},{{x1,x2,...}}] | the Jacobian for a vector function f |
Vector derivatives.
This gives the gradient of the function
.
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This gives the Jacobian for a vector function.
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does in fact depend on 
, you can use the explicit functional form 
. "The Representation of Derivatives" describes how objects like 
work. 
