TITLE: Is the derivative of a big-O class the same as the big-O class of the derivative?
QUESTION [8 upvotes]: Basically, for every function $f(x) \in O(g(x))$, is $f'(x) \in O(g'(x))$?

REPLY [11 votes]: No, consider $f(x)=x$ and $g(x)=\sin(x^3)$. Then, certainly $g\in O(f)$, but $f'(x)=1$, and $g'(x)=3x^2\cos(x^3)$ and $\displaystyle \limsup_{x\to\infty} 3x^2\cos(x^3)=\infty$.