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 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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If y does in fact depend on x, you can use the explicit functional form y[x]. "The Representation of Derivatives" describes how objects like y'[x] work.
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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 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 Hessian.
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This gives the Jacobian for a vector function.
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