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

Chain rule

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

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

and

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

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

Then also the composition

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

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

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