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.

Proof sketch: Write linear approximations with remainders: $f(a+h)=f(a)+Df(a)h+r_f(h),\quad \frac{\|r_f(h)\|}{\|h\|}\to 0,$ and similarly for $g$ at $f(a)$. Substitute the first into the second and control the remainder terms using continuity of $Dg$ at $f(a)$ (or directly from differentiability), yielding the stated derivative .