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

Remarks

In matrix form, the Jacobian matrix of $f \circ g$ at $a$ is the product $J_f(g(a)) \, J_g(a)$. In terms of partial derivatives, $\dfrac{\partial (f \circ g)_i}{\partial x_j}(a) = \sum_{l=1}^m \dfrac{\partial f_i}{\partial y_l}(g(a)) \, \dfrac{\partial g_l}{\partial x_j}(a)$. The same rule, with the same proof, holds for Fréchet derivatives of maps between normed spaces. A special case is the chain rule for a composition with a curve.

Explanations

L1 human

The derivative of a composition is the composition of the derivatives: first the linear approximation of $g$, then the linear approximation of $f$. In one variable this is $(f \circ g)'(a) = f'(g(a)) \, g'(a)$, a product of two numbers. In several variables the numbers become matrices.