Answer
$c = \sqrt(kRT)$
Work Step by Step
We have the isentropic solution $PV^k = Constant = P/(\rho ^k)$
differentiating the above equation with respect to$ \rho $
$\frac{ ( \delta P . \rho^k - Pk\rho^(k-1)\delta \rho)}{\rho^(2k)}$ $ \delta\rho = 0$ ,
$\delta P = \frac{Pk}{\rho} \delta \rho$
$\frac{\delta P}{ \delta \rho} = \frac{Pk}{\rho} = \frac{\rho RTk}{\rho} = kRT$
let speed of sound = c
We know
$c^2$ = $\frac{\delta P}{ \delta \rho} = kRT$
so $c = \sqrt(kRT)$
