Chain rule

Chain rule: Let $U\subseteq\mathbb R^k$ and $V\subseteq\mathbb R^m$ be open sets. Let $f:U\to V$ be differentiable at $a\in U$, and let $g:V\to\mathbb R^p$ be differentiable at $f(a)$. Then the composition $g\circ f$ is differentiable at $a$, and

In terms of the Jacobian matrix, this becomes the matrix identity $J_{g\circ f}(a)=J_g(f(a))\,J_f(a)$, which is the standard “multiply Jacobians” rule.