Theorem. chain rule [br915]

Suppose $E$ is an open set in $\mathbb{R}^{n}$, $\mathbf{f}$ maps $E$ into $\mathbb{R}^{m}$, $\mathbf{f}$ is differentiable at $\mathbf{x}_{0} \in E$, $\mathbf{g}$ maps an open set containing $\mathbf{f}(E)$ into $\mathbb{R}^{k}$, and $\mathbf{g}$ is differentiable at $\mathbf{f}(\mathbf{x}_{0})$. Then the mapping $\mathbf{F}$ of $E$ into $\mathbb{R}^{k}$ defined by $$ \mathbf{F}(\mathbf{x}) = \mathbf{g}\left( \mathbf{f}(\mathbf{x}) \right) $$ is differentiable at $\mathbf{x}_{0}$, and $$ \mathbf{F}'(\mathbf{x}_{0}) = \mathbf{g}'\left( \mathbf{f}(\mathbf{x}_{0}) \right) \mathbf{f}'(\mathbf{x}_{0}). $$