= Solution
If $F:X\to Y$ is a nonconstant <holomorphic map> between compact connected <Riemann surfaces>, its <local degree of a holomorphic map>[local degree] at $p$ is the integer $m_F(p)\geq1$ for which suitable local coordinates give
$$
F(z)-F(p)=az^{m_F(p)}+O(z^{m_F(p)+1}),
\qquad a\ne0.
$$
The <valency theorem> states that
$$
\sum_{p\in F^{-1}(y)}m_F(p)
$$
is independent of $y\in Y$. This common value is the <degree of a holomorphic map>, denoted $\deg F$.
Now let $f$ be a nonconstant <rational function> of degree $d$. If its distinct finite <poles> have orders $m_1,\ldots,m_r$, and its pole order at infinity is $m_\infty\geq0$, then
$$
d=m_1+\cdots+m_r+m_\infty.
$$
The <derivative> $f'$ has a pole of order $m_j+1$ at each finite pole. When $m_\infty>0$, the expansion $f(z)\sim cz^{m_\infty}$ shows that $f'(z)\sim cm_\infty z^{m_\infty-1}$ at infinity. Hence the <degree of the derivative of a rational function> is
$$
\deg f'=
\begin{cases}
d+r-1,&m_\infty>0,\\
d+r,&m_\infty=0.
\end{cases}
$$
In the first case $r\geq0$, while in the second $1\leq r\leq d$; therefore
$$
\boxed{d-1\leq\deg f'\leq2d.}
$$
For every $d\geq1$, the lower bound is attained by $f(z)=z^d$, whose derivative has degree $d-1$ (with a constant assigned degree zero). For distinct $a_1,\ldots,a_d$, the function
$$
f(z)=\sum_{j=1}^d\frac1{z-a_j}
$$
has $d$ simple poles, degree $d$, and a derivative with $d$ double poles, so $\deg f'=2d$. Thus both rational bounds are sharp for every $d$.
Let next $g$ be a nonconstant <elliptic function> for the <period lattice> $\Lambda$. Its <degree of an elliptic function>[degree] is the total order of its poles in a <fundamental parallelogram>; by the valency theorem, this is also the degree of the induced map $\mathbb C/\Lambda\to\widehat{\mathbb C}$. If these poles have orders $n_1,\ldots,n_s$, then $d=\deg g=\sum_jn_j$, and $g'$ has poles of orders $n_j+1$. The <degree of the derivative of an elliptic function> is consequently
$$
\deg g'=d+s.
$$
Since every nonconstant elliptic function has at least one pole and $1\leq s\leq d$,
$$
\boxed{d+1\leq\deg g'\leq2d.}
$$
Let $d\geq3$ be odd. The <Weierstrass elliptic function> supplies the lower-bound example
$$
g(z)=\wp(z)^{(d-3)/2}\wp'(z).
$$
It has one pole modulo $\Lambda$, of order $d$, so its derivative has one pole of order $d+1$. For the upper bound, choose distinct points $a_1,\ldots,a_d$ modulo $\Lambda$ and nonzero constants $c_j$ with $\sum_jc_j=0$. The quasi-periodicity of the <Weierstrass zeta function> makes
$$
g(z)=\sum_{j=1}^dc_j\zeta(z-a_j)
$$
elliptic. It has exactly $d$ simple poles, while $g'$ has $d$ double poles. Therefore the two bounds are attained for every required odd degree.
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