Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

Given $x = t^2$ and $y = t^3$.

To find $\frac{dy}{dx}$ for parametric equations:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

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Finding $\frac{dy}{dt}$:

$y = t^3$

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

Finding $\frac{dx}{dt}$:

$x = t^2$

$\frac{dx}{dt} = 2t$

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Therefore:

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

$\frac{dy}{dx} = \frac{3t}{2}$

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To find $\frac{d^2y}{dx^2}$:

$\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \times \frac{dt}{dx}$

Finding $\frac{d}{dt}\left(\frac{3t}{2}\right)$:

$\frac{d}{dt}\left(\frac{3t}{2}\right) = \frac{3}{2}$

Finding $\frac{dt}{dx}$:

$\frac{dt}{dx} = \frac{1}{dx/dt}$

$\frac{dt}{dx} = \frac{1}{2t}$

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Therefore:

$\frac{d^2y}{dx^2} = \frac{3}{2} \times \frac{1}{2t}$

$\frac{d^2y}{dx^2} = \frac{3}{4t}$