Extended Mean-Value Theorem

Let the functions $f$ and $g$ be Differentiable on the Open Interval $(a,b)$ and Continuous on the Closed Interval $[a,b]$. If $g'(x) \not = 0$ for any $x \in (a,b)$, then there is at least one point $c \in(a,b)$ such that

$${f'(c)\over g'(c)} = {f(b)-f(a)\over g(b)-g(a)}.$$

See also Mean-Value Theorem