Calculus/Multivariable Calculus/Chain Rule

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Rules of taking Jacobians[edit | edit source]

If f : RmRn, and h(x) : RmR are differentiable at 'p':

Important: make sure the order is right - matrix multiplication is not commutative!

Chain rule[edit | edit source]

The chain rule for functions of several variables is as follows. For f : RmRn and g : RnRp, and g o f differentiable at p, then the Jacobian is given by

Again, we have matrix multiplication, so one must preserve this exact order. Compositions in one order may be defined, but not necessarily in the other way.