Question

If $x=t^2$, $y=t^3$, then $\frac{d^{2}y}{d{x}^2}$ equals to ?

• A. $\frac{3}{2}$
• B. $\frac{3}{4t}$
• C. $\frac{3}{2t}$
• D. $\frac{3t}{2}$

Answer 1

The correct option is B

$\frac{3}{4t}$

Explanation of the correct answer:

It is given that, $x=t^2$ and $y=t^3$.

$\frac{dx}{dt}=2t$

Now,

$$\begin{gathered} \frac{dy}{dt}=3t^2 \\ \Rightarrow \frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}} \\ \Rightarrow \frac{dy}{dx}=\frac{3t^2}{2t}\left[\because \frac{dx}{dt}=2t\right] \\ \Rightarrow \frac{dy}{dx}=\frac{3t}{2} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{3}{2}\times \frac{dt}{dx} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{3}{2}\times \frac{1}{2t} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{3}{4t} \end{gathered}$$

Hence, Option $\left(\mathrm{B}\right)$ is correct.