Answer
See explanation
Work Step by Step
The definition for the volume expansivity is $$
\beta=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P
$$ The definition for the isothermal compressibility is $$
\alpha=-\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T
$$ According to the cyclic relation, $$
\left(\frac{\partial v}{\partial T}\right)_P\left(\frac{\partial P}{\partial v}\right)_T\left(\frac{\partial T}{\partial P}\right)_v=-1
$$ which on rearrangement becomes $$
\left(\frac{\partial v}{\partial T}\right)_P=-\left(\frac{\partial v}{\partial P}\right)_T\left(\frac{\partial P}{\partial T}\right)_v
$$ When this is substituted into the definition of the volume expansivity, the result is $$
\begin{aligned}
\beta & =-\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T\left(\frac{\partial P}{\partial T}\right)_v \\
& =-\alpha\left(\frac{\partial P}{\partial T}\right)_v
\end{aligned}
$$
