If $y=2at, x=at^2, a≠0, $ then $\frac{d^2y}{dx^2}$ is equal to :

$-\frac{a}{xy}$

$y=2at, x=at^2$

$\frac{dy}{dt}=2at$

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

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

$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{1}{t}\right)$

$⇒\frac{d}{dt}\left(\frac{1}{t}\right)\frac{dt}{dx}$

$⇒\frac{-1}{t^2}.\frac{1}{\frac{dx}{dt}}$

$⇒\frac{-1}{t^2}.\frac{1}{2at}$

$⇒\frac{-1}{2at^3}$