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