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Let $f(x)$ be a real valued and continuously differentiable function. Suppose that $f(x) = \mathcal{O}(g(x))$ in the sense that there exists a positive real number $M$ and a real number $x_0$ such that $|f(x)| \leq M g(x)$ for all $x \geq x_0$. Let $f'(x) = d/d x \ f(x)$. Then, what are the additional requirements needed to impose on $f(x)$ to conclude that $f'(x) = \mathcal{O}(g(x))$?

user304347
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    There are possibly many conditions, one of which is that $f'(x)\in \mathcal{O}\Big( f(x) \Big)$. Do you have a particular condition in mind? – Keen-ameteur Mar 14 '21 at 12:19
  • Thanks for the comment! I'm hoping to find conditions on the "smoothness" of $f(x)$ but I'd very curious to see what are other possible conditions. – user304347 Mar 14 '21 at 13:18

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