Calculus/Multivariable Calculus/Chain Rule

Rules of taking Jacobians

If f : R^m → Rⁿ, and h(x) : R^m → R are differentiable at 'p':

• $J_{𝐩}(𝐟+𝐠)=J_{𝐩}𝐟+J_{𝐩}𝐠$
• $J_{𝐩}(h𝐟)=h{J}_{𝐩}𝐟+𝐟(𝐩)J_{𝐩}h$
• $J_{𝐩}(𝐟\cdot 𝐠)={𝐠}^T{J}_{𝐩}𝐟+{𝐟}^T{J}_{𝐩}𝐠$

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

Chain rule

The chain rule for functions of several variables is as follows. For f : R^m → Rⁿ and g : Rⁿ → R^p, and g o f differentiable at p, then the Jacobian is given by

$\left(J_{𝐟(𝐩)}𝐠\right)\left(J_{𝐩}𝐟\right)$

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.

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