Derivative of a pointwise limit of a sequence of functions

Question

It is easy to construct a sequence of differentiable functions $f_n(x)$ converging pointwise to a function that is not differentiable (a simple example is $\tan^{-1}(nx)$). And it is a theorem that if in addition the derivatives $f_n'(x)$ converge uniformly (on an interval $[a,b]$ for example), then the limit function is differentiable with derivative the limit of the derivatives. I am looking for an example (if any exist) of a sequence of differentiable functions $f_n(x)$ that converges pointwise to a differentiable function $f(x)$, and such that the sequence of derivatives $f_n'(x)$ also converges pointwise to a differentiable function $g(x)$, but $f'\neq g$. If such an example does not exist, I would like to see the proof of the corresponding theorem.