In an exercise, I am asked to show that
If in the (single variable) Inverse Function Theorem $f$ has $k$ continuous derivatives, then the inverse function $g$ also has $k$ continuous derivatives.
Other posts on this forum answered a similar question using Leibniz's formula, but I have not yet learned this. I suspect that I can apply Taylor's theorem to write $f$ as its $k$th Taylor polynomial plus its remainder, since $f$ is $C^k$, but I am struggling to figure out what I should do next. Could someone please help me with how to proceed?
Thank you