If $x=at^4$ and $y=2at^2$, then $\frac{d^2y}{dx^2}$ is equal to:

$-\frac{1}{2at^6}$

The correct answer is Option (4) → $-\frac{1}{2at^6}$

$x=at^4$ and $y=2at^2$

$⇒\frac{dx}{dt}=4at^3$    ...(1)

$⇒\frac{dy}{dt}=4at$    ...(2)

from (1) and (2),

$⇒\frac{dy}{dx}=\frac{1}{t^2}$

$⇒\frac{d^2y}{dx^2}=\frac{-2}{t^3}×\frac{dt}{dx}$

$=\frac{-2}{t^3}×\frac{1}{4at^3}$

$=\frac{-1}{2at^6}$