Question

Find $\frac{d^{2}y}{d{x}^2}$, of the following functions: $x=a{t}^2,y=2at$.

Answer 1

Given,

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

Then $\frac{dx}{dt}=2at$......(1) and $\frac{dy}{dt}=2a$.......(2).

Now,

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{1}{t}$.[ Using (1) and (2)]

Now,

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

or,$\frac{d^{2}y}{d{x}^2}=-\frac{1}{t^2}.\frac{1}{2at}=-\frac{1}{2a{t}^3}$ [ Using (1)]