= Degree of the derivative of a rational function
{title2=$\deg f'$}
Let a nonconstant rational function $f$ have degree $d$, let $r$ be its number of distinct finite poles, and let $m_\infty$ be its pole order at infinity, taken as zero when infinity is not a pole. Differentiation raises each finite pole order by one and changes a pole of order $m_\infty>0$ at infinity into one of order $m_\infty-1$. Hence
$$
\deg f'=
\begin{cases}
d+r-1,&m_\infty>0,\\
d+r,&m_\infty=0.
\end{cases}
$$
In particular, $d-1\leq\deg f'\leq2d$.
Back to article page
