Do the property of $f_n$ carry over to $f_n'$?

Question

Suppose a sequence of differentiable functions $f _ { n } : \mathbb { R } \rightarrow [ 0,1 ]$ converges pointwise to the zero function. Does it follow that the derivatives $f _ { n } ^ { \prime }$ converge pointwise to the zero function?

Answer 1

No! Consider $$f_n(x)=\frac{\sin nx}{\sqrt{n}}$$ on $\Bbb R$. Here each $f_n$ is infinitely differentiable and $f_n$ converges pointwise (actually uniformly) to zero. But $$f_n'(x)=\sqrt{n} \cos nx$$ is divergent at each $x \in \Bbb R$