Computing degree of a map

Question

Define $f:\mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ by $$f([z_0,z_1])=[p(z_0,z_1),z_1^n],$$ where $p(z_0,z_1)=z_0^n+c_{n-1}z_0^{n-1}z_1+ \dots c_1z_0 z_1^{n-1}+c_0 z_1^n$ is an arbitrary homogeneous polynomial. I would like to compute the degree of $f$. I think I need to use the following: for a regular point $x \in \mathbb{CP}^1$, the differential $df_x:T_x\mathbb{CP}^1 \rightarrow T_{f(x)}\mathbb{CP}^1$ is orientation preserving.
I am not entirely sure how to show this. I know that $df_x$ is orientation preserving if its Jacobian is positive. However, can I compute it in charts? For instance, in the chart $U_0=\{[z_0,z_1] \mid z_0 \neq 0\}=\{[1,z] \mid z\in \mathbb{C}\}$, we have $$f([1,z])=[p(1,z),z^n],$$ whose Jacobian is $0$.