When does $f\sim g$ implies $f'\sim g'$?

Question

Given two $C^1$ functions $f,g:[0,+\infty)\to [0,+\infty)$ such that $f(x)\sim g(x)$ as $x\to\infty$, which good conditions guarantee that $f'(x)\sim g'(x)$?

I thought that monotonicity of the derivatives could be such condition, but I wasn't able to prove or disprove. I think that one such condition can be $f$, $f'$, $g$, $g'$, $f-g$ and $f'-g'$ to be monotonic increasing, but that's so restrictive that hurts.

Also, if anyone could suggest a book that treats problems like that (I don't know, "asymptotic analysis"?) I would appreciate. Thanks in advance.

Answer 1

Monotonicity of the derivatives is not enough. Let $f(x)=1-\frac{1}{x+1}$ and $g(x)=1-\frac{1}{(x+1)^2}$.

Answer 2

For solutions where $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}g(x)$ are finite, I believe a good solution may be $(f(x)-\lim\limits_{x\to\infty}f(x))\sim(g(x)-\lim\limits_{x\to\infty}g(x))$