TITLE: Show there exists a unique $f$ (in $\mathbb R^+$) such that $\frac{d}{dx}f(x)=f^{-1}(x)$
QUESTION [7 upvotes]: Question: Show there exists a unique bijection $f:\mathbb R^+\to\mathbb R^+$ such that $\frac{d}{dx}f(x)=f^{-1}(x)$, where the right-hand side is the functional inverse.
I figured I would start by finding a trivial example of existence, but 1) I can't think of one and 2) I don't know how I'd prove uniqueness.

REPLY [7 votes]: For an example, try $f$ of the form $f(x) = c x^p$.  See what $c$ and $p$ must be to make this work.