If \(x = t^{1/2}, y = t^{3/2}\), then \(\displaystyle\frac{dy}{dx} =\)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(3t\)
For a curve given in parametric form, \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\):
\( \displaystyle \frac{dx}{dt} = \frac{1}{2}t^{-1/2},\quad \frac{dy}{dt} = \frac{3}{2}t^{1/2} \)
\( \displaystyle \frac{dy}{dx} = \frac{\frac{3}{2}t^{1/2}}{\frac{1}{2}t^{-1/2}} \)
\(= 3t^{1/2 + 1/2} = 3t\)
Shortcut: Since \(x^3 = t^{3/2} = y\), the curve is \(y = x^3\). So \(\frac{dy}{dx} = 3x^2 = 3t\).