Let $f:[a,b]\to\mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$ Define the function $$ h(x)=f(x)\left(g(b)-g(a)\right)-g(x)\left(f(b)-f(a)\right)-f(a)g(b)+f(b)g(a). $$ Because $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$ so is $h$ Furthermore, $h(a)=h(b)=0$ so by Rolle's theorem there exists a $\xi\in(a,b)$ such that $h'(\xi)=0$ This implies that $$ f'(\xi)\left(g(b)-g(a)\right)-g'(\xi)\left(f(b)-f(a)\right)=0 $$ and, if $g(b)\neq g(a)$ $$ \frac{f'(\xi)}{g'(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)}. $$