Exercise with differential in coordinates

Question

Let $x,y,z,t$ be the standard coordinates for $\mathbb R^4$, and let $$f: S^3 \to \mathbb {CP}^1, \;\; f (x,y,z,t)=[x+iy:z+it].$$ I must show that $f $ is $C^\infty $ and find $df_p $ in coordinates; it is immediate that $f $ is $C^\infty $, because the inclusion of $S^3$ in $\mathbb R ^4/\{0\}\cong\mathbb C^2/\{0\}$ and the projection of $\mathbb C^2/\{0\}$ over $\mathbb {CP}^1$ are both $C^\infty $. However I am very confused about what "in coordinates" means; although I read several times my notes (and the page of Wikipedia) I couldn't apply the theoretical notions even to a simple problem. I think that this exercise could be explanatory; can you tell me a way to resolve it? Thanks a lot