Differentiable function/D open in K/Chain rule/Fact

Chain rule

Let ${}D,E\subseteq {\mathbb {K} }$ denote open subsets and let

f\colon D\longrightarrow {\mathbb {K} }

and

g\colon E\longrightarrow {\mathbb {K} }

be functions with ${}f(D)\subseteq E$ . Suppose that ${}f$ is differentiable in ${}a$ and that ${}g$ is differentiable in ${}b:=f(a)$ .

Then also the

g\circ f\colon D\longrightarrow {\mathbb {K} }

is differentiable in ${}a$ and its derivative is

{}(g\circ f)'(a)=g'(f(a))\cdot f'(a)\,.