Assume that $f$ is an arbitrary monotonically increasing on $[0,1]$. I am trying to compute the following:
$\frac{d}{dx}f^{-1}(f(x)-1)$
Where $f^{-1}$ is the inverse function.
The current best line of work I have is using the chain rule and inverse function rules I get:
$f'(x) \cdot 1/f'[f^{-1}(f(x)-1)]$
However, this seems to just be creating an infinitely nested function. As such I am stuck.
Thanks!