I'm trying to solve the following problem (which seems to be the same as in Derivative of a pointwise limit of a sequence of functions but it does not have answers)
If $f_n:[a,b] \to \mathbb{R}$ is a sequence of continuously differentiable functions on $[a,b]$ that converges pointwise to a continuously differentiable function $f$ on $[a,b]$ and if the sequence $(f'_n)$ converges pointwise to a function $g$ on $[a,b]$, does it follow that $f'=g$?
I know that if $(f'_n)$ converges uniformly on $[a,b]$ to a function $g$ then $g=f'$, but I'm not sure if that happens if $(f'_n)$ just converges pointwise. I'm having trouble finding a counterexample.