Chain rule in several variables

Statement

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open sets, let $g \colon U \to \mathbb{R}^m$ with $g(U) \subseteq W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$. Suppose that $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$. Then the composition $f \circ g \colon U \to \mathbb{R}^k$ is differentiable at $a$, and $$D(f \circ g)(a) = Df(g(a)) \circ Dg(a).$$