Chain Rule for $C^1$ Maps Between Euclidean Spaces

Let $n,m,p\in \mathbb{N}$. Let $U\subseteq \mathbb{R}^n$, $V\subseteq \mathbb{R}^m$, and $W\subseteq \mathbb{R}^p$ be open subsets. Let $f:U\to V$ and $g:V\to W$ be $C^1$ maps. Then the composition $g\circ f:U\to W$ is again of class $C^1$. Moreover, for every $a\in U$, the Jacobian matrices from the differentiability definition satisfy

where the product on the right-hand side is the matrix product from the definition of matrix multiplication.