03 - Chain Rule

Chain Rule

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $g:{\mathbb{R}}^m\to {\mathbb{R}}^p$, $\underline{a}\in {\mathbb{R}}^m$

Assume that $f$ is differentiable at $\underline{a}$ and $g$ is differentiable at $f(\underline{a})$

Let $h:=g\circ f$ then

$$h\text{ is differentiable at }\underline{a}\text{ with }dh(\underline{a})=dg(f(\underline{a}))df(\underline{a})$$

Omitted + Non-Examinable