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Basically, for every function $f(x) \in O(g(x))$, is $f'(x) \in O(g'(x))$?

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The important thing is that $O$ cares about global behavior, how fast the function grows as $x \to \infty$. The derivative cares about local behavior, and can be very large if the function is highly oscillatory, without the function increasing rapidly. –  Ross Millikan Dec 18 '12 at 0:05
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up vote 10 down vote accepted

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 $\displaystyle \limsup_{x\to\infty} 3x^2\cos(x^3)=\infty$.

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Okay, so the follow-up here is, what are the classes of functions called where this is true? I'm pretty sure polynomials and exponentials do this. –  Joe Z. Dec 9 '12 at 19:19
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