# Derivative of a exponentially decaying smooth function

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?

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.