A sequence of differentiable functions can converge uniformly to a differentiable function \(f\) while the derivatives converge pointwise to a function \(g\neq f'\).
For \(n\geq 1\) let \(f_n\colon\RR\to\RR\), \(f_n(x) \coloneqq \frac{x}{1+nx^2}\). Since \(1+nx^2\geq 2\sqrt{n}\abs{x}\), we have \(\abs{f_n(x)}\leq \frac{1}{2\sqrt n}\) for all \(x\in\RR\), with equality at \(x = 1/\sqrt n\). Hence \(\norm{f_n}_\infty = \frac{1}{2\sqrt n}\to 0\), and \(f_n\to f\coloneqq 0\) uniformly on \(\RR\).
The derivatives are \[ f_n'(x) = \frac{1-nx^2}{(1+nx^2)^2}. \] For every \(n\), \(f_n'(0) = 1\); for \(x\neq 0\), \(\abs{f_n'(x)}\leq \frac{1+nx^2}{(1+nx^2)^2} = \frac{1}{1+nx^2}\to 0\). Hence \(f_n'\to g\coloneqq\chi_{\theset{0}}\) pointwise on \(\RR\), while \(f' = 0\) and \(g(0) = 1\neq f'(0)\).