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(D) For an adiabatic process,the relation between pressure $P$ and volume $V$ is given by:$P V^{\gamma} = 	ext{constant}$Taking the derivative with r

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$\frac{5 P_0}{2 V_0}$

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(D) For an adiabatic process,the relation between pressure $P$ and volume $V$ is given by:$P V^{\gamma} = 	ext{constant}$Taking the derivative with respect to $V$:$\frac{d}{dV}(P V^{\gamma}) = 0$$V^{\gamma} \frac{dP}{dV} + P \cdot \gamma V^{\gamma-1} = 0$$\frac{dP}{dV} = -\gamma \frac{P}{V}$The magnitude of the rate of change of pressure with volume is $\left| \frac{dP}{dV} ight| = \gamma \frac{P}{V}$.For an ideal monatomic gas,the adiabatic index $\gamma = \frac{5}{3}$.At point $A$,the pressure $P = 3 P_0$ and the volume $V = 2 V_0$.Substituting these values:$\left| \frac{dP}{dV} ight| = \frac{5}{3} 	imes \frac{3 P_0}{2 V_0} = \frac{5 P_0}{2 V_0}$.Thus,the correct option is $D$.

The variation of pressure $P$ with volume $V$ for an ideal monatomic gas during an adiabatic process is shown in the figure. At point $A$,the magnitude of the rate of change of pressure with respect to volume is

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