Question:
If $x=2 a t, y=a t^{2}$, where $a$ is a constant, then find $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{2}$.
Solution:
Given:
$x=2 a t, y=a t^{2}$
$\frac{d x}{d t}=2 a ; \frac{d y}{d t}=2 a t$
$\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
$=\mathrm{t}$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx}}{\mathrm{dt}}}$
$=\frac{1}{2 a}$