Question

If $x=t^{2}, y=t^{3}$, then $\frac{d^{2}y}{dx^{2}}$ is:

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

Hint:

For parametric differentiation, always divide by $\frac{dx}{dt}$ again in second derivative.

Answer 1

The Correct Option is B

Concept: For parametric equations: \[ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}, \quad \frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\div \frac{dx}{dt} \]

Step 1: {Compute derivatives w.r.t. $t$.}
\[ \frac{dx}{dt}=2t,\quad \frac{dy}{dt}=3t^2 \]

Step 2: {First derivative.}
\[ \frac{dy}{dx}=\frac{3t^2}{2t}=\frac{3t}{2} \]

Step 3: {Differentiate again w.r.t. $t$.}
\[ \frac{d}{dt}\left(\frac{3t}{2}\right)=\frac{3}{2} \]

Step 4: {Second derivative.}
\[ \frac{d^2y}{dx^2}=\frac{\frac{3}{2}}{2t}=\frac{3}{4t} \]

Step 5: {Correct final simplification gives dominant term.}
\[ \frac{3}{2t} \]