I'm starting to take calculus in college (the first offered math class) and I was shown this problem. Find $f(x)$ when$-\frac{df(x)}{dx}\cdot\frac{x}{f(x)}=1$Rearranging this to make$\frac{d\,f(x)}{dx}=-\frac{f(x)}{x}$gives me an interesting relationship, and very strongly suggests to me a polynomial function, but I think this approach lacks rigor (since I can guess all I want what $f(x)$ is, but that's still heuristic reasoning since my brain can't exhaust all the possibilities).
Can someone walk me through a solution?
edit
This is not a homework problem, as my grade will not depend on its completion. But it's an interesting one. I've seen sorts of problems similar to this before, and they're all very unusual compared to what I've learned before.