Degree bounds for the derivative of a rational function

For a nonconstant rational function of degree $n$, its derivative has degree between $n-1$ and $2n$. A finite pole of order $m$ becomes one of order $m+1$. If infinity is a pole of order $p-q>0$, it contributes $p-q-1$ to the derivative's pole count, where $p,q$ are the degrees of coprime numerator and denominator. This proves the lower bound; the quotient rule denominator $Q^2$ proves the upper bound. The functions $z^n$ and $1/(z^n-1)$ attain the bounds.