Big-Theta Notation. Is this theorem true? Is the following sentence true assuming that $f$ and $g$ are differentiable and their derivatives are continous? I'd say yes, but don't know how to show it.
$$g(x) \in  \Theta(f(x)) \iff \frac{d}{dx} g(x) \in \Theta\left(\frac{d}{dx}f(x)\right)$$

At least one direction fails. Take the function $f(x)=x$, and modify it by adding a bump function that, for every positive integer $n$, is $0$ between $n$ and $n+e^{-n}$, climbs to $1$ at $n+e^{-2n}$, and falls back to $0$ at $n+e^{-3n}$. All derivatives exist, but the first derivative is very wild for large $x$.    
