If one has a uniformly convergent sequence of differentiable functions $f_n$ (say from $\mathbb{R}$ to $\mathbb{R}$), we know that the limit $f$ is not always differentiable. Even if it is differentiable, the derivative $f'(x)$ is not necessarily the limit of the derivatives $f'_n(x)$.
But in all the counter-examples I know, there is a sequence $x_n$ (with the limit $x$) such that $f'(x)$ is the limit of $f'_n(x_n)$.
My question is : Is it always true ? I have tried to use Rolle's theorem, but did not succeed. Or do you know also a counter-example for this ? Thank you.