Let $S^2 \subseteq \mathbb{R}^3$ be the unit sphere in $\mathbb{R}^3$ (which is a smooth manifold of dimension $2$). Let $\phi = (x_1, x_2): S^2 \backslash \{N\} \to \mathbb{R}^2 $ be the stereographic projection of the $S^n$, where $N = e_3 = \pmatrix{0 \\ 0 \\1}$ is the north pole, i.e. $\phi$ is given by
$$\phi(u) = \phi(u_1, u_2, u_3) = \left( \frac{u_1}{1 - u_3}, \frac{u_2}{1 - u_3} \right) $$
Now let $f: \mathbb{R}^3 \to \mathbb{R}$ be given by $f(u) = u_3$, i.e. $f$ is the third coordinate projection.
For any $p \in S^2$, I now want to determine the derivation $\frac{\partial}{\partial x_i} \mid_p (\overline{f})$ (where $\overline{f}$ denotes the function germ of $f$).
I must admit that I haven't yet fully understood how to determine a derivation like this. So do I simply need to determine the partial derivatives of $f$, and then determine $\phi(p)$ and sum up the partial derivatives of $f$ correctly to get the derivative of $f$ in the "direction" $\phi(p)$? I'm not really sure if that's the right approach, though.
