We are given:
$\displaystyle PV=mRT$
hence:
$\displaystyle P=\frac{mRT}{V}\,\therefore\,\frac{\delta P}{\delta V}=-\frac{mRT}{V^2}$
$\displaystyle V=\frac{mRT}{P}\,\therefore\,\frac{\delta V}{\delta T}=\frac{mR}{P}$
$\displaystyle T=\frac{PV}{mR}\,\therefore\,\frac{\delta T}{\delta P}=\frac{V}{mR}$
and so:
$\displaystyle \frac{\delta P}{\delta V}\frac{\delta V}{\delta T}\frac{\delta T}{\delta P}=\left(-\frac{mRT}{V^2} \right)\left(\frac{mR}{P} \right)\left(\frac{V}{mR} \right)=-\frac{mRT}{PV}=-\frac{PV}{PV}=-1$