When does two functions being almost equal imply their derivatives are also almost equal

Question

I am sorry if this question is vague. Consider two functions $f(x)$ and $g(x)$. Their values are almost equal at all points. But their derivatives need not be equal. For example, let $f(x)=0$ and $g(x)=\sin(10^{15}x)\times10^{-5}$. Then $|f(x)-g(x)| \leq 10^{-5}$ $\forall x$ but $|\frac{df(x)}{dx}-\frac{dg(x)}{dx}|$ could be as large as $10^{10}$. My question is given $|f(x)-g(x)| \lt \delta$ what minimum additional information about these functions does one need that allows to bound $|\frac{df(x)}{dx}-\frac{dg(x)}{dx}|$.