If $x = t^{1/2} ,y = t^{3/2}$, then $\frac{dy}{dx}=$

$3t$

The correct answer is Option (1) → $3t$

$x = t^{\frac{1}{2}}, \ y = t^{\frac{3}{2}}$

$\frac{dx}{dt} = \frac{1}{2} t^{-\frac{1}{2}}$

$\frac{dy}{dt} = \frac{3}{2} t^{\frac{1}{2}}$

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\frac{3}{2} t^{\frac{1}{2}}}{\frac{1}{2} t^{-\frac{1}{2}}}$

$= \frac{3}{2} t^{\frac{1}{2}} \times \frac{2}{1} t^{\frac{1}{2}} = 3 t^{1} = 3t$