Question

If $x=a{t}^2,y=2at$, then $\frac{d^{2}x}{d{y}^2}=$ ________.

• A. $\frac{-1}{2a{t}^3}$
• B. $-2a{t}^3$
• C. $\frac{-1}{t^2}$
• D. $\frac{1}{2a}$

Answer 1

The correct option is A $\frac{-1}{2a{t}^3}$

Given $x=a{t}^2$ and $y=2at$

differentiating above equations w.r.to $t$

we get

$\frac{dx}{dt}=2at$ and $\frac{dy}{dt}=2a$

by chain rule,

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{2a}{2at}$

$⟹\frac{dy}{dx}=\frac{1}{t}$

differentiating $\frac{dy}{dx}$ w.r.t to $x$ we get

$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$

$=\frac{d}{dt}\left(\frac{dy}{dx}\right).\frac{dt}{dx}$ (chain rule (multiplying and dividing by $dt$))

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

$=\frac{-\frac{1}{t^2}}{2at}=-\frac{1}{2a{t}^3}$