Question:
Write the correct alternative in the following:
If $x=t^{2}, y=t^{3}$, then $\frac{d^{2} y}{d x^{2}}=$
A. $\frac{3}{2}$
B. $\frac{3}{4 t}$
C. $\frac{3}{2 t}$
D. $\frac{3 t}{2}$
Solution:
Given:
$x=t^{2} ; y=t^{3}$
$\frac{\mathrm{dy}}{\mathrm{dt}}=3 \mathrm{t}^{2} ; \frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{t}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{3 \mathrm{t}}{2}$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{\frac{3 \mathrm{t}}{2}}{2 \mathrm{t}}$
$=\frac{3}{4}$