Explanation

(d) $P = F\ v = (ma)v = m\left( \frac{d^{2}x}{dt^{2}} \right)\left( \frac{dx}{dt} \right)$
Since, power is constant
$\left( \frac{d^{2}x}{dt^{2}} \right)\left( \frac{dx}{dt} \right) = k$
or $\frac{d}{dt}\left( \frac{dx}{dt} \right)^{2} = k$
$\left( \frac{dx}{dt} \right)^{2} = k_{1}t$
$\frac{dx}{dt} = \sqrt{k_{1}t}$
$\frac{dx}{dt} = k_{2}(t)^{1/2}\ \ \ \ (\therefore k_{1}^{1/2} = k_{2})$
$x = k_{3}t^{3/2}\ \ \left( \because k_{3} = \frac{2}{3}k_{2} \right)$
Hence $\frac{dx}{dt} \propto t^{1/2} \propto x^{1/3}$