Suggestion for a functional equation $f'\left(\frac{a}{x}\right)=\frac{x}{f(x)}$ where $f:(0,\infty)\to(0,\infty)$ is differentiable I want to solve the following functional equation:

Find all differentiable functions $f : (0,\infty) \rightarrow (0,\infty)$ for which there is a positive real number $a$ such that $$f'\left(\frac{a}{x}\right)=\frac{x}{f(x)}$$ for all $x > 0$.

I have noted that the function $f$ is increasing but i cannot go any further
Any suggestions?

It seems that $f \approx x^\beta $ might work.
