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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 $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)} = 0$.

Syd Amerikaner
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    I would say no (or you will need very strong conditions on $f$ and $g$). For example, $f(x)=x,\sin\left(\dfrac{1}{x}\right),g(x)$. Then, $f(x)\in o\big(g(x)\big)$. However, $$\dfrac{f'(x)}{g'(x)}=\sin\left(\dfrac1x\right),\left(\frac{g(x)}{g'(x)}+x\right)-\frac{\cos\left(\frac1x\right)}{x},\left(\frac{g(x)}{g'(x)}\right),.$$ If $\lim\limits_{x\to 0},\dfrac{g(x)}{g'(x)}$ exists and is nonzero, then $\dfrac{f'(x)}{g'(x)}$ is not bounded. That is, for most nicely behaved $g$, there is always a counterexample $f$ to $f'(x)\in o\big(g'(x)\big)$. – Batominovski Apr 25 '20 at 13:40
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    @Batominovski has given a good answer in a comment, but I'd like to describe the same idea in words, which might make it easier to see where it came from. A function $f$ can have all its values very small and yet have very large slope (derivative) at some points, just by oscillating up and down very rapidly. In particular, $f$ can be way smaller than $g$ while $f'$ is (at some points) way bigger than $g'$. – Andreas Blass Apr 25 '20 at 13:47
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    In addition to the two excellent comments above, you do have something of the sort for analytic functions. If $$ f(x) = \sum_{k \geq i} a_k x^k, \quad g(x) = \sum_{k \geq j} b_k x^k, $$ with $a_i, b_j \neq 0$ and $j \geq 1$, then saying $f(x) = o(g(x))$ means $i > j$. For the same reason, you then have $f'(x) = o(g'(x))$. But well, that is a lot of assumptions for something not too exciting, and as mentioned, wrong more often than not. – Raoul Apr 25 '20 at 15:07
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    thanks for the excellent comments! As I said, I am aware of the $\sin(1/x)x$ counter example but I came across the assertation (that derivative and little o can be interchanged) in a paper I have to replicate for my thesis so I had to make sure that I am not missing a point here. Thank you guys! – Syd Amerikaner Apr 27 '20 at 01:32
  • Thanks for the replies! – Syd Amerikaner May 01 '20 at 00:57

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