Prove the inverse of this differentiable function is differentiable? Suppose we have a differentiable function $ g $ that maps from a real interval $ I $ to the real numbers and suppose $ g'(r)>0$ for all $ r$ in $ I $. 
Then I want to show that $ g^{-1}$ is differentiable on $g(I). $
Intuitively this makes sense but I can't come up with a neat proof. I was thinking to use the mean value theorem but I'm not sure if that would get me anywhere.

If $g'(r)>0$ then $g(r)$ in injective. 
Exist $g^{-1}(r)$.
Then use the chain rule for derivate $g^{-1}(g(r))=r$.
