Write the correct alternative in the following:

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:

$x=t^{2} ; y=t^{3}$

$\frac{\mathrm{dy}}{\mathrm{dt}}=3 \mathrm{t}^{2} ; \frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{t}$

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{3 \mathrm{t}}{2}$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{\frac{3 \mathrm{t}}{2}}{2 \mathrm{t}}$

$=\frac{3}{4}$

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