Disprove: If $ f_n , f $ are differentiable and $ f_n \to f $ uniformly, then $ f_n' \to f' $ ( pointwise )

Question

Prove\Disprove: If $ f_n , f $ are differentiable and $ f_n \to f $ uniformly, then $ f_n' \to f' $ ( pointwise ).

I was told the theorem is false but I couldn't come up with an example. Can you please help? how would you find an example that satisfies all the assumptions in question like this, is there some sort of rule-of-thumb I don't know about?

Answer 1

Try $f_n(x) = \frac{sin {nx}}{n}$ with $f(x) = 0$

Answer 2

A slight variation of a previous counterexample! $f_{n}(x)=x^{2}\dfrac{sinnx}{n}$. It is clear that $f_{n}(x)\to 0$ pointwise and $sup_{[0,1]}|f_{n}(x)-0|\to 0$. But $f_{n}'(x)=2x\dfrac{sinnx}{n}+x^{2}cosnx.n.\dfrac{1}{n}$=

=$2x\dfrac{sinnx}{n}+x^{2}cosnx$ and clearly $f_{n}'(x)$ does not converge to $f(x)=0$.