= Solution
An <elliptic function> for a <lattice> $\Lambda$ is a <meromorphic function> on $\mathbb C$ satisfying $f(z+\lambda)=f(z)$ for all $\lambda\in\Lambda$. Its order is the number of poles, counted with multiplicity, in a fundamental parallelogram, equivalently the degree of its map from the <torus> to the <Riemann sphere>. Differentiation preserves the periods. If the distinct poles have orders $r_1,\ldots,r_s$, the <derivative> has exactly the same pole locations with orders $r_j+1$. Therefore
$$
\boxed{n=m+s,\qquad m+1\leq n\leq2m.}
$$
The bounds use $1\leq s\leq m$, valid for a nonconstant <elliptic function>.
For the <Weierstrass elliptic function> $\wp$, $f=\wp\wp'$ has a single pole of order five, so its <derivative> has order six. For the second example choose $c$ not a critical value of $\wp$ and choose $a$ away from zero and the two points where $\wp=c$. Then
$$
f(z)=\wp(z-a)+\frac{\wp'(z)}{\wp(z)-c}
$$
has a double pole at $a$ and three simple poles: the quotient has residue $-2$ at zero and residue 1 at each of its two other poles. No locations overlap. Hence its order is five and its <derivative> has order $3+2+2+2=9$, as required.
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