Is there any relation between the convergence of a differentiable function and the convergence of its derivative?
I understand that whether $f'$ converges to zero does not tell us anything about the convergence of $f$ as there exist functions like $f_1$, \begin{equation} f_1(x) = \frac{\sin x^3}{x}, \quad f_{1}^{'}(x)=3x\cos x^3 -\frac{\sin x^3}{x^2} \end{equation} where $f_1 \to 0$ but $f_{1}^{'}$ grows unbounded and oscillates rapidly as $x\to\infty$. And there are also functions like $f_2$, \begin{equation} f_2(x) = \ln x, \quad f_{2}^{'}(x) = \frac{1}{x} \end{equation} where $f_2 \to \infty$ but $f_{2}^{'} \to 0$ as $x\to \infty$.
Therefore, I am curious about if there exist some additional conditions (maybe something like the Barbalat lemma?) that allow us to establish a link between the convergence of a function and the convergence of its derivative.