# 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 :
 In:= Out= This gives the third derivative:
 In:= Out= You can differentiate with respect to any expression that does not involve explicit mathematical operations:
 In:= Out= D does partial differentiation. It assumes here that y is independent of x:
 In:= Out= 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:
 In:= Out= 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:
 In:= Out= 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 :
 In:= Out= This gives the Hessian:
 In:= Out= This gives the Jacobian for a vector function:
 In:= Out= 