The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class.
But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ in any interval containing zero, but for the inverse $f^{-1}(x)= x^{1/3}$ the derivative is not defined in $0$. Am I missing something? Is there a hypothesis missing?
You have to assume that $f'(x) \neq 0$ for all $x$, I think. You have that $$ \left(f^{-1}\right)'(f(x)) = \frac{1}{f'(x)} \text{,} $$ and if that is to hold for all $x$, then neither $(f^{-1})'$ nor $f'$ can have any zeros.