Differentiable function/D in R/Chain rule/Fact/Proof

Proof

Due to fact, one can write

${}f(x)=f(a)+f'(a)(x-a)+r(x)(x-a)\,$

and

${}g(y)=g(b)+g'(b)(y-b)+s(y)(y-b)\,.$

Therefore,

${}{\begin{aligned}g(f(x))&=g(f(a))+g'(f(a))(f(x)-f(a))+s(f(x))(f(x)-f(a))\\&=g(f(a))+g'(f(a)){\left(f'(a)(x-a)+r(x)(x-a)\right)}+s(f(x)){\left(f'(a)(x-a)+r(x)(x-a)\right)}\\&=g(f(a))+g'(f(a))f'(a)(x-a)+{\left(g'(f(a))r(x)+s(f(x))(f'(a)+r(x))\right)}(x-a).\end{aligned}}$

The remainder function

${}t(x):=g'(f(a))r(x)+s(f(x))(f'(a)+r(x))\,$

is continuous in ${}a$ with value ${}0$.

To fact