Differentiation
| D[f,x] | partial derivative  |
| D[f,x,y,...] | multiple derivative  |
| D[f{x,n}] | nth derivative  |
| D[f,x,NonConstants->{v1,v2,...}] |
|  with the vi 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 y is independent of x.
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Instead of giving an explicit function y[x], you can tell D that y implicitly depends on x. D[y, x, NonConstants->{y}] then represents  , with y implicitly depending on x.
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| D[f,{{x1,x2,...}}] | the gradient of a scalar function f ( f/x1, f/x2, ... ) |
| D[f,{{x1,x2,...},2}] | the Hessian matrix for f |
| D[f,{{x1,x2,...},n}] | the nth-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 x2+y2.
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This gives the Jacobian for a vector function.
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