Check of uniform convergence of a sequence of function

Question

The given sequence of functions is $$ f_n = -\frac{e^{-x^2n^2}}{n} $$

1. Prove that $f_n$ tends to $0$ uniformly;

2. Prove that $f_n'$ tends to $0$ pointwise but not uniformly.

I have tried by using some conventional rule as Weierstrass' $M$-test, but falied to prove.

Answer 1

Hint for 1). The pointwise limit is $0$ and $|f_n(x)|=\frac{e^{-n^2x^2}}{n}$ attains it maximum value at $0$ (why?).

Hint for 2). Note that $$f_n'(x)=2nxe^{-n^2x^2}.$$ The pointwise limit is again $0$, but note that $f'(1/n)=\frac{2}{e}$. What may we conclude?