If \(x=e^t\) and \(y = e^{2t}\) then \(\displaystyle\frac{d^2y}{dx^2}=\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
In parametric form, both are differentiated with respect to t:
\( \displaystyle \frac{dx}{dt} = e^t,\quad \frac{dy}{dt} = 2e^{2t} \)
\( \displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2e^{2t}}{e^t} \)
\(= 2e^t\)
Differentiating again with respect to x, using the chain rule through t:
\( \displaystyle \frac{d^2y}{dx^2} = \frac{d}{dt}\left(2e^t\right) \times \frac{dt}{dx} \)
\( \displaystyle = 2e^t \times \frac{1}{e^t} = 2 \)
Shortcut: eliminating t, since \(y = e^{2t} = (e^t)^2\),
\(y = x^2\)
\(y' = 2x\)
\(y'' = 2\)