$\mathbb{C} \to \mathbb{C}, z \to z^n$ is of degree n and its derivative of degree $n-1$. But in another chart, the function is $u \to \frac{1}{u^n}$ and its derivative is $\frac{-n}{u^{n+1}}$, of degree n+1.
What am I getting wrong here? Note: I use "degree" as in "degree of the covering of $X \to f(X)$"