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

This tests the second-order derivative of a parametric function — a common exam trap.

Concept:

• For parametric $x(t)$, $y(t)$: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.
• Crucially, $\frac{d^2y}{dx^2} = \frac{d}{dt}(dy/dx) \div (dx/dt)$ — you must divide again by $dx/dt$, NOT differentiate $dy/dx$ with respect to t alone.

Why B is correct:

• $\frac{dy}{dx} = \frac{3}{2}t$; differentiating w.r.t. t gives 3/2; dividing by $dx/dt = 2t$ gives $\frac{3}{4t}$.

Why the distractors are wrong:

• A (3/2): stops at $\frac{d}{dt}(dy/dx)$ and forgets to divide by $dx/dt = 2t$.
• C ($\frac{1}{2t^2}$) and D ($\frac{3}{2t}$): come from mis-dividing or omitting the factor 2 from dx/dt.