Question

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

Answer 1

Given,

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

On differentiating both sides w.r.t. t, we get,

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

Therefore,

$\frac{dy}{dx}=\frac{2a}{2at}=\frac{1}{t}$

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

$=\frac{d}{dt}\left(\frac{1}{t}\right)\times \frac{1}{2at}=-\frac{1}{t^2}\times \frac{1}{2at}=-\frac{1}{2a{t}^3}$