Showing multivariable maps are smooth How does one in practice actually show a map is smooth? Say I am considering: $$F:B^n\to \Bbb R^n,\qquad F(x) = \frac{x}{\sqrt{1-\|x\|^2}},$$ where $B^n$ is the open unit ball in $\Bbb R^n$ and $\|x\|$ is the euclidean norm on $\Bbb R^n$. Smoothness means that all partial derivatives of all orders exist. I can compute the Jacobian matrix, which seems quite ugly, but even still, I can't just deduce that the partial derivatives of all orders exist (and they don't vanish in few steps). Usually the maps I encounter are obviously smooth, but only because they are polynomial in each entry and what-not, but outside of these cases, I don't know how to approach this. Are there some results that I haven't seen? How do you actually show such a map is smooth?

Your function is smooth because it can be obtained from smooth functions ($x$, square root, and $\lVert x\rVert^2$) using composition and arithmetic opeations.