If $x=e^t$ and $y = e^{2t}$ then $\frac{d^2y}{dx^2}=$

2

The correct answer is Option (3) → 2

Given:

$x = e^{t}$ and $y = e^{2t}$

$\frac{dy}{dt} = 2e^{2t}$

$\frac{dx}{dt} = e^{t}$

Then,

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2e^{2t}}{e^{t}} = 2e^{t}$

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

= $\frac{d}{dt}(2e^{t}) \times \frac{dt}{dx}$

$= 2e^{t} \times \frac{1}{\frac{dx}{dt}} = 2e^{t} \times \frac{1}{e^{t}} = 2$

Therefore, $\displaystyle \frac{d^2y}{dx^2} = 2$.