I know that it is in general not true that if $f(x) = o(x^n)$ it follows that $f'(x) = o(x^{n-1})$ or in other words: $f(x) = o(g(x))$ does not imply $f'(x) = o(g'(x))$. But is it still possible to relate those two quantities together? E.g. by bounding it or making additional assumptions about the convergence (that is, when does $f(x) = o(g(x))$ imply $f'(x) = o(g'(x))$)?
By definition of $o$ we have $f(x) = o(g(x))$ iff $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)} = 0$.