Theorem (3.22)

If $\mathcal{U}\subset \mathbb{R}^{n}$ and $\mathcal{V}\subset \mathbb{R}^{m}$ be Open sets and let $f:\mathcal{U}\to \mathcal{V}$ and $g:\mathcal{V}\to \mathbb{R}^{l}$ be maps of Class $C^{r}$. Then the composition $g\circ f:\mathcal{U}\to \mathbb{R}^{l}$ is of class $C^{r}$, and its derivative is given by $$\boldsymbol D(g\circ f)(\boldsymbol x)=\boldsymbol Dg(f(\boldsymbol x))\circ \boldsymbol Df(\boldsymbol x)\in \mathscr{L}(\mathbb{R}^{n};\mathbb{R}^{l})$$