Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be even smooth function. Let $w(x)=f(\sqrt x)$ for $x>0$. On MO https://mathoverflow.net/questions/65264 I found the following integral formula $w^{(k)}(x^2)=\frac{(2x)^{-2k+1}}{(k-1)!} \int_0^x (x^2-t^2)^{k-1} f^{(2k)}(t) dt.$
How to proof it and show that there exist limits $\lim_{x\rightarrow 0} w^{(k)}(x^2)$ for $k=1,2,\ldots$? By it would be follow smoothness of $w$ on $[0,\infty)$.