Second derivative of a parametric equation

The second derivative is $\frac{d^2y}{dx^2} = \frac{ \dot{x}\ddot{y} - \dot{y} \ddot{x} } {\dot{x}^3}$

Given:

$\dot{x} = 2t^2$

$\dot{y} = 3t^4$

Differentiate each with respect to $t.$

$\ddot{x} = 4t$

$\ddot{y} = 12t^3$

Substitute these into the equation above.

$\frac{d^2y}{dx^2} = \frac{(2t^2)(12t^3) - (3t^4)(4t)}{(2t^2)^3} = \frac{24t^5 - 12t^5}{8t^6} = \frac{12t^5}{8t^6} = \frac{3}{2t}$

Plug in $t=\frac32.$

$\frac{d^2y}{dx^2} = \frac{3}{2({\frac32})} = 1$