Chain Rule

Statements

Theorem. (@Boo75 Theorem 2.3) Let $U\subset {\mathbb{R}}^n$ and $V\subset {\mathbb{R}}^m$ be open sets, $F:U\to V$ be a vector-valued function differentiable at $a\in U$ and $G:V\to {\mathbb{R}}^p$ be a vector-valued function differentiable at $b=F(a)\in V$. Then, $H=G\circ F$ is differentiable at $x=a$ and we have $DH(a)=DG(F(a))DF(a).$