Problem in solving a question related to derivative.

Question

The question is :

If $f : \mathbb R \longrightarrow \mathbb R$ is differentiable and bijective.Then is $f^{-1}$ differentiable?

It is clear that here $f$ is either strictly increasing or strictly decreasing.But from here how can I proceed to prove or disprove the above result.Please give me a hint.Then I will try.

Thank you in advance.

Answer 1

This is a false result.

Take for instance

$$f:x\mapsto x^3.$$

Then $f^{-1}(x)=\sqrt[3]{x}$ which is not differentiable at $x=0$ because

$$\lim_{x\to 0} \frac{\mathrm d}{\mathrm d x}(f^{-1})(x)=+\infty.$$

Answer 2

Hint

$(f^{-1})'(y)$ exists $\iff f'(f^{-1}(y))\neq 0$.