Derivative and uniform convergence

Question

$f_n(x) = \dfrac{\arctan (n^{1/4} x^2)}{n^{3/2}}$ I need to calculate first derivative of it and then tell if first derivative is uniformly convergent. I calculated it but I got now idea how to bound it using Weierstrass thesis to show it is uniform convergent. Help please! :)

Answer 1

A standard computation shows

$$ f_n'(x) = \frac{2x}{n^{5/4}(1+ n^{1/2}x^4)} \implies |f_n'(x)| \le \frac{2|x|}{n^{5/4}(1+ x^4)}.$$

Since $2|x|/(1+x^4)|$ is bounded on $\mathbb R,$ there is a constant $C$ such that for each $n,$ $|f_n'(x)| \le C/n^{5/4}$ for all $x\in \mathbb R.$ It follows that $f_n\to 0$ uniformly on $\mathbb R.$