A uniformly convergent sequence of differentiable functions whose derivatives do not converge pointwise

For \(n\geq 1\) let \(f_n\colon\RR\to\RR\), \(f_n(x) \coloneqq \frac{\sin(nx)}{n}\). Then \(\abs{f_n(x)}\leq\frac1n\) for all \(x\), so \(f_n\to 0\) uniformly on \(\RR\). Each \(f_n\) is differentiable with \(f_n'(x) = \cos(nx)\), and \(f_n'(\pi) = \cos(n\pi) = (-1)^n\), so the sequence \((f_n'(\pi))\) diverges and \((f_n')\) does not converge pointwise on \(\RR\).