Differentiable function/Mean value theorem/Quotient version/Fact

Mean value theorem (quotient version)

Let $b>a$, and suppose that

$f,g:[a,b]⟶ℝ$

are continuous functions which are differentiable on $]a,b[$ and such that

$g^{\prime }(x)\ne 0$

for all $x\in ]a,b[$.

Then

$g(b)\ne g(a)$, and there exists some $c\in ]a,b[$ such that

$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime }(c)}{g^{\prime }(c)}.$

Proof, Write another proof