This is only a half-thought out question right now, and I'll probably answer it myself. But I'm posting it as I came up with it so that, after I work on it, I can check on here and find out how other people approached it.
Okay. So there is a way to find the derivative of a function if you know the derivative of its inverse, like so:
$g'(x) = \frac{1}{f'(g(x))}$ where $f$ is the inverse of $g$.
Now let's say that I know $g$, $f'$ and $g'$ but I don't know $f$. If I have:
$f'(g(x))g'(x)=1$
Can I solve for f?
If so, how?