Answer
$\left(\frac{\partial Z}{\partial T}\right)_P=0$
Work Step by Step
The inversion curve is the locus of the points at which the Joule-Thompson coefficient $\mu$ is zero, $$
\mu=\frac{1}{c_P}\left(T\left(\frac{\partial v}{\partial T}\right)_P-v\right)=0
$$ which can also be written as $$
T\left(\frac{\partial v}{\partial T}\right)_P-\frac{Z R T}{P}=0\tag{a}
$$ since it is given that $$
v=\frac{Z R T}{P}\tag{b}
$$ Taking the derivative of $(b)$ with respect to $T$ holding $P$ constant gives $$
\left(\frac{\partial v}{\partial T}\right)_P=\left(\frac{\partial(Z R T / P)}{\partial T}\right)_P=\frac{R}{P}\left(T\left(\frac{\partial Z}{\partial T}\right)_P+Z\right)
$$ Substituting in (a), $$
\begin{aligned}
\frac{T R}{P}\left(T\left(\frac{\partial Z}{\partial T}\right)_P+Z\right)-\frac{Z R T}{P} & =0 \\
T\left(\frac{\partial Z}{\partial T}\right)_P+Z-Z & =0 \\
\left(\frac{\partial Z}{\partial T}\right)_P & =0
\end{aligned}
$$ which is the desired relation.
