Proof of Inverse Function Theorem, class k

Question

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?

Answer 1

Yes, the Inverse Function Theorem, which tells you precisely that $$ D(F^{-1})=(DF\circ F^{-1})^{-1} $$

Answer 2

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.