# Proof of Inverse Function Theorem, class k

I was studying the Inverse Function Theorem, and I found this proof on the internet:

http://virtualmath1.stanford.edu/~andras/174A-2.pdf

In the proof, there is this line about $$C^k$$ functions:

If $$F$$ is $$C^k$$, $$k > 1$$, then $$DF$$ is $$C^{k−1}$$, hence $$(DF)^{−1}$$ is $$C^{k−1}$$, hence $$F^{-1}$$ is $$C^k$$.

Now what I don't get is the last "hence" part, since $$(DF)^{-1}\neq D(F^{-1})$$. Is there any reasoning in why this is true?

Yes, the Inverse Function Theorem, which tells you precisely that $$D(F^{-1})=(DF\circ F^{-1})^{-1}$$
You can express the derivative $$D (F^{-1})$$ of $$F^{-1}$$ in terms of $$F$$ and the derivative of $$F$$. If these are both differentiable then so is $$D (F^{-1})$$, just because it can be expressed through the composition of differentiable maps. The general case is just by induction.