Let $$ \varphi:\mathbb{R}^2 \longrightarrow \mathbb{S}^2-{N} \subset \mathbb{R}^3$$ be the (inverse) stereographic projection from the North pole on the unit sphere centred at the origin.
$$\varphi(x,y) = \left( \frac{2x}{x^2+y^2+1},\frac{2y}{x^2+y^2+1},\frac{x^2+y^2-1} {x^2+y^2+1} \right) $$ Let $$\hat{f}(x,y) = (\alpha x, \alpha y) $$ where $\alpha >0. $ $$f: \mathbb{S}^2 \longrightarrow \mathbb{S}^2 = \left\{ \begin{array}{ll} \varphi \circ \hat{f} \circ \varphi^{-1} & on \quad \mathbb{R}^2 \longrightarrow \mathbb{S}^2-{N} \\ f(N)=N & \\ \end{array} \right. $$
How would I show that the map is smooth at $N$ and compute $Df|_N$ at the north pole?
In the Jacobian representation, $Df|_N$ goes to infinity.
I tried taking $$\lim_{\epsilon=(a,b,c)\rightarrow (0,0,0)} \frac{f(N+\epsilon)-f(N)}{\epsilon} $$ But the same problem happens.
Is there a reformulation of $f$, say $h$ which is equal to $f$ but expressed entirely differently (for example in polar coordinates?)
Any hints or clues are appreciated!