If x = t², y = t³, then d²y/dx²

The correct option is 3/4t.

To find the second derivative of $y$ with respect to $x$, denoted as $\frac{d^{2}y}{d{x}^2}$, when $x$ and $y$ are given in terms of a parameter $t$, we use parametric differentiation.

First, we are given the parametric equations:
$x=t^2$
$y=t^3$

Step 1: Find the first derivatives with respect to the parameter $t$.
Differentiating $x$ with respect to $t$:
$\frac{dx}{dt}=\frac{d}{dt}\left(t^2\right)=2t$
Differentiating $y$ with respect to $t$:
$\frac{dy}{dt}=\frac{d}{dt}\left(t^3\right)=3t^2$

Step 2: Find the first derivative of $y$ with respect to $x$, $\frac{dy}{dx}$.
Using the chain rule, we have:
$\frac{dy}{dx}=\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}=\frac{3t^2}{2t}=\frac{3}{2}t$

Step 3: Find the second derivative, $\frac{d^{2}y}{d{x}^2}$.
The second derivative is the derivative of $\frac{dy}{dx}$ with respect to $x$:
$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$
Since $\frac{dy}{dx}$ is expressed in terms of the parameter $t$, we must apply the chain rule again:
$\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}$

Substitute $\frac{dy}{dx}=\frac{3}{2}t$ and $\frac{dx}{dt}=2t$ into the equation:
$\frac{d}{dt}\left(\frac{3}{2}t\right)=\frac{3}{2}$
Now divide this result by $\frac{dx}{dt}$:
$\frac{d^{2}y}{d{x}^2}=\frac{\left(\frac{3}{2}\right)}{2t}=\frac{3}{4t}$

Thus, the second derivative $\frac{d^{2}y}{d{x}^2}$ is equal to 3/4t.