If $f(x)\ll1$ is it safe to assume that $f^{\prime}(x)\ll1$?

Question

If $$f(x)\ll1$$ is it safe to assume that $$f^{\prime}(x)\ll1$$

Answer 1

No, consider something like

$$ f(x) = e^{-x}\sin(e^{x}) $$ $$ f'(x) = -e^{-x}\sin(e^{x}) + e^{-x}\cos(e^{x})e^{x} = \cos(e^{x}) - e^{-x}\sin(e^{x}) $$ Then $f(x)$ gets small very quickly, but $f'(x)$ asymptotically oscillates faster and faster between $-1$ and $1$.

Answer 2

No. You could have a function like $e^{-x^2}\sin(e^{x^2})$, which is very small for large values of $x$, but whose derivative can be very large.

Answer 3

No. Consider $f(x)=\frac1x\sin x^3$ as $x\gg 0$.

Answer 4

Another example, maybe simpler : $$f_\epsilon(x)=\epsilon \sin(\epsilon^{-2} x)$$

It will oscillate at a speed of $\epsilon^2$ with an amplitude of $\epsilon$, and therefore will have a derivative of order $\frac{1}{\epsilon}$.

EDIT : thanks to Ian, I corrected an error