Answer
The proof is below.
Work Step by Step
For an isothermal process, the following is true:
$PV=c$
Where c is an arbitrary constant. Thus, we find:
$\frac{dP}{dV}=-\frac{p}{V}$
For an adiabatic process, the following is true:
$PV^{\gamma}=c$
Where c is an arbitrary constant. Thus, we find:
$ \frac{dP}{dV}=-\gamma \frac{P}{V}$
Thus, the proof is complete.
