If t=e^2 x and y=log _e t^2, then (d^2 y)/(d x^2) is:

The correct answer is Option (1) → 0

$t=e^{2 x}$, $y=\log _e t^2$

so $\log t=2x$

so $\frac{\log t}{2}=x,y=2\log t$

so $\frac{dx}{dt}=\frac{1}{2t}$, $\frac{dy}{dt}=\frac{2}{t}$

$\frac{dy}{dx}=4⇒\frac{d^2y}{dx^2}=0$