Answer
$\dfrac{\partial P}{\partial V}\dfrac{\partial V}{\partial T}\dfrac{\partial T}{\partial P}=-1$
Work Step by Step
Recall some relationships such as; $P= \dfrac{mRT}{V} ,V=\dfrac{mRT}{P}; T= \dfrac{PV}{mR}$
For $\dfrac{\partial P}{\partial T}$, all other variables aside from $T$ to be treated as constants.
and $\dfrac{\partial P}{\partial T}=\dfrac{\partial}{\partial T}[\dfrac{mRT}{V}]=\dfrac{mR}{V}$
Here, we have $\dfrac{\partial T}{\partial P}=\dfrac{V}{mR}$
and $\dfrac{\partial P}{\partial V}\dfrac{\partial V}{\partial T}\dfrac{\partial T}{\partial P}=\dfrac{-mRT}{V^{2}}\times \dfrac{mR}{P}\times\dfrac{V}{mR}$
and $\dfrac{-mRT}{PV}=\dfrac{-mRT}{\dfrac{mRT}{V}\times V}=-1$
Hence, we have $\dfrac{\partial P}{\partial V}\dfrac{\partial V}{\partial T}\dfrac{\partial T}{\partial P}=-1$
