Differentiable function/D in R/Chain rule/Fact

Chain rule

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

f\colon D\longrightarrow \mathbb {R}

and

g\colon E\longrightarrow \mathbb {R}

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 {R}

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

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