Let $U \subset \mathbb{R}^m$ and $V \subset \mathbb{R}^k$ be open sets. Let $g:U \to V$ be differentiable at $a \in U$, and let $f:V \to \mathbb{R}^n$ be differentiable at $g(a) \in V$. Write $g=(g_1,\ldots,g_k)$, where each component map $g_r:U \to \mathbb{R}$ is defined by $g(x)=(g_1(x),\ldots,g_k(x))$. Then, for every $i \in \{1,\ldots,m\}$,

\begin{align*} \partial_{x_i}(f\circ g)(a)=\sum_{r=1}^k \partial_{y_r}f(g(a))\,\partial_{x_i}g_r(a). \end{align*}

Here $\partial_{y_r}f(g(a)) \in \mathbb{R}^n$ is the [partial derivative](/page/Partial%20Derivative) of the vector-valued map $f$ in the $r$-th coordinate direction of $\mathbb{R}^k$, and the product $\partial_{y_r}f(g(a))\,\partial_{x_i}g_r(a)$ denotes scalar multiplication in $\mathbb{R}^n$.