If $x = t^2$ and $y = t^3$, then $\frac{d^2y}{dx^2}$ is equal to

$\frac{3}{4t}$

The correct answer is Option (2) → $\frac{3}{4t}$ ##

We have, $x = t^2$ and $y = t^3$

On differentiating w.r.t. $t$ both Eqs., we get

$\frac{dx}{dt} = 2t \text{ and } \frac{dy}{dt} = 3t^2$

$∴\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3}{2}t$

On further, differentiating w.r.t. $x$, we get

$\frac{d^2y}{dx^2} = \frac{3}{2} \frac{d}{dt}t \cdot \frac{dt}{dx}$

$= \frac{3}{2} \cdot \frac{1}{2t} \quad \left[ ∵\frac{dt}{dx} = \frac{1}{2t} \right]$

$= \frac{3}{4t}$