Chain Rule for Differentiable Maps Between Euclidean Spaces

Let $n$, $m$ and $p$ be natural numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $V$ be an open subset of $\mathbb{R}^m$. Let $f:U\to\mathbb{R}^m$ satisfy $f(x)\in V$ for every $x\in U$, let $g:V\to\mathbb{R}^p$, and let $g\circ f:U\to\mathbb{R}^p$ denote the map defined by $(g\circ f)(x)=g(f(x))$.

Let $a\in U$, let $A$ be a real matrix with $m$ rows and $n$ columns, let $B$ be a real matrix with $p$ rows and $m$ columns, and write $BA$ for the product of real matrices, a real matrix with $p$ rows and $n$ columns.

Suppose that $f$ is differentiable at $a$ with derivative matrix $A$, and that $g$ is differentiable at $f(a)$ with derivative matrix $B$. Then $g\circ f$ is differentiable at $a$ with derivative matrix $BA$.

Consequently, by claim 2 of A Derivative Matrix is the Jacobian Matrix, and is Unique, the Jacobian matrices $Df(a)$, $Dg(f(a))$ and $D(g\circ f)(a)$ are all defined, and

$$D(g\circ f)(a)=Dg\bigl(f(a)\bigr)\,Df(a).$$