Answer
$T\dfrac{\partial P}{\partial T} \dfrac{\partial V}{\partial T}=mR$
Work Step by Step
Since, $P= \dfrac{mRT}{V} ,V=\dfrac{mRT}{P}; T= \dfrac{PV}{mR}$
$\dfrac{\partial P}{\partial T}=\dfrac{\partial}{\partial T}[\dfrac{mRT}{V}]=\dfrac{mR}{V}$ ...(1)
$\dfrac{\partial V}{\partial T}=\dfrac{\partial}{\partial T}[\dfrac{mRT}{P}]=\dfrac{mR}{P}$ ....(2)
From the equatiosn (1) and (2) , we get
$T \times \dfrac{\partial P}{\partial T} \times \dfrac{\partial V}{\partial T}=(T) \times \dfrac{mR}{V} \times \dfrac{mR}{P}$
Hence, $T \times \dfrac{\partial P}{\partial T} \times \dfrac{\partial V}{\partial T}=(\dfrac{PV}{mR}) (\dfrac{mR}{V}) (\dfrac{mR}{P})=mR$
