Setting $S^{n} := \{x\in\mathbb{R}^{n+1}: \|x\| = 1\}$, and labelling the north and south poles as $N:= (0,\ldots,0,1)$, $S:=(0,\ldots,0,-1)$, I can set the coordinate charts up as follows:
Let $U_N = S^n - N$ and $U_S = S^n - S$.
Taking the usual stereographical projections $\phi_{N}:U_{N}\to\mathbb{R}^{n}$ and $\phi_S : U_S \to\mathbb{R}^n$, we turn $S^n$ into a topological manifold of dimension $n$.\ \ I'm having trouble verifying that this structure satisfies the definition of a differentiable manifold, as I do not know how to check that $f:=\phi_{N}\circ \phi_{S}^{-1}:\mathbb{R}^{n} - 0$ is $C^{\infty}$. It is clear how to show (once I draw the picture) that $f$ is a bijection.
Do I need to derive a formula for the $\phi_{N}$ and $\phi_{S}^{-1}$ map in order to show that it is $C^{\infty}$? Any references I found this in usually display it as an example and mention casually that it is easy to show that $f$ is $C^\infty$, but I'm not even sure how to begin.