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Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function such that $|f(x)|=O(e^{-c|x|})$ for some $c>0$. Is it possible to infer something about the decay of its derivative?

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Not much, no. For instance, $f(x) = \sin(\exp(x^2)) \exp(-x) = O(\exp(-x))$ but $\lim \sup_{x \to \infty} |f'(x)| = +\infty$. In general, you can choose $g$ where $g$ is smooth and very fast growing (faster than $\exp(x)$) so that $f(x) = \sin(g(x)) \exp(-x)$ admits a derivative with arbitrarily bad behaviour at infinity.

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