Question

\[ \text{If } x = 2 \cos^3 \theta \text{ and } y = 3 \sin^2 \theta, \text{ then } \frac{dy}{dx} =\ ? \]

• A. \( -\sec \theta \)
• B. \( \cos \theta \)
• C. \( -\csc \theta \)
• D. \( \sin \theta \)

Hint:

When both \( x \) and \( y \) are given in terms of a third variable (like \( \theta \)), use parametric differentiation: \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \).

Answer 1

The Correct Option is A

We are given: \[ x = 2 \cos^3 \theta, y = 3 \sin^2 \theta \] Differentiate both \( x \) and \( y \) with respect to \( \theta \): \[ \frac{dx}{d\theta} = 2 \cdot 3 \cos^2 \theta \cdot (-\sin \theta) = -6 \cos^2 \theta \sin \theta \] \[ \frac{dy}{d\theta} = 3 \cdot 2 \sin \theta \cos \theta = 6 \sin \theta \cos \theta \] Now use the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{6 \sin \theta \cos \theta}{-6 \cos^2 \theta \sin \theta} \] Cancel out common terms: \[ \frac{dy}{dx} = \frac{1}{- \cos \theta} = -\sec \theta \] % Final Answer \[ \boxed{-\sec \theta} \]