Uniform convergence of derivative of sequence of functions

Question

Consider $f_n(x)=n^{-1}\exp(-n^2x^2)$, sequence of functions from $R$ to $R$. I wanted to check the convergence of derivative of this function. Derivative is given by $f'_n=-2nx\exp(-n^2x^2)$. As n tends to infinity, exponential function dominates and the function converges to $0,$ so $f'_n$ converges pointwise. Now inorder to check about uniform convergence, supremum norm tends to 0 as n tends to infinity. But, I'm not quite sure about the uniform convergence near origin. Is my approach correct? Is $f'_n$ uniformly convergent? Or something has to be taken care near origin. Thanks in advance.

Answer 1

$f_n'(\frac 1n)=-2e^{-1}$ so $f_n'$ cannot converge uniformly to $0$.

[Note that $\sup_x |f_n'(x)| \geq f_n'(\frac 1n)$ and that if $(f_n')$ converges uniformly the limit can only be $0$ because of pointwise convergence to $0$]