Chain rule (multivariable)

Chain rule (multivariable): Let $U\subseteq\mathbb{R}^n$ and $V\subseteq\mathbb{R}^m$ be open. Suppose $f:U\to V$ is differentiable at $a\in U$ and $g:V\to\mathbb{R}^p$ is differentiable at $f(a)$. Then $g\circ f:U\to\mathbb{R}^p$ is differentiable at $a$ and $D(g\circ f)(a)=Dg(f(a))\circ Df(a).$ In matrix form (with Jacobians ), $J_{g\circ f}(a)=J_g(f(a))\,J_f(a).$

The chain rule is the main computational law of multivariable differentiation and underlies coordinate changes, implicit differentiation, and optimization.