Assume that $f:I: \rightarrow J$ is a bijection of class $C^1$ and $f'(x)>0$ for all $x \in I$, where $I$, $J$ are intervals in $\mathbb R$. Then by the known theorem $f^{-1}$ is of class $C^1$ and $(f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}$ for $y\in J$.
Assume now that $f$ is of class $C^n$, $n \in \mathbb N$. Why $f^{-1}$ is also of class $C^n$?