If $ x=2t^3 $ and $y = 3t^4 ,$ then $\frac{d^2y}{dx^2}$ is equal to :
Answer & explanation
Correct answer: option 2
Given
$x=2t^{3},\; y=3t^{4}$
$\frac{dx}{dt}=6t^{2},\;\frac{dy}{dt}=12t^{3}$
$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{12t^{3}}{6t^{2}}=2t$
$\frac{d}{dt}\left(\frac{dy}{dx}\right)=2$
$\frac{d^{2}y}{dx^{2}}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}=\frac{2}{6t^{2}}$
$\frac{d^{2}y}{dx^{2}}=\frac{1}{3t^{2}}$
final answer: $\frac{1}{3t^{2}}$