If x=3 at^2, y=3 at^4 then (d y)/(d x) is:

The correct answer is Option (2) - $2 t^2$

$x = 3at^2,\;\; y = 3at^4$

$\frac{dx}{dt} = 6at$

$\frac{dy}{dt} = 12at^3$

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

$= \frac{12at^3}{6at} = 2t^2$

$t^2 = \frac{x}{3a}$

$\frac{dy}{dx} = 2 \cdot \frac{x}{3a} = \frac{2x}{3a}$

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