Cauchy mean value theorem

Cauchy mean value theorem: Let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Assume $g'(x)\neq 0$ for all $x\in(a,b)$ and $g(b)\neq g(a)$. Then there exists $c\in(a,b)$ such that

Taking $g(x)=x$ recovers the mean value theorem . This theorem is the main tool behind L'Hôpital's rule for indeterminate limits.