Question

Evaluate the derivative of $ y = \cos x \times \sin y, \quad \frac{dy}{dx} \text{ at } \left( \frac{\pi}{6}, \frac{\pi}{5} \right) $

• A. \( -\frac{1}{2} \)
• B. \( 0 \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)

Hint:

When working with implicit differentiation, be sure to apply the chain rule and product rule carefully.

Answer 1

The Correct Option is A

We are given the equation: \[ y = \cos x \times \sin y \] To find \( \frac{dy}{dx} \), we differentiate both sides of the equation implicitly with respect to \( x \): \[ \frac{d}{dx} \left( y \right) = \frac{d}{dx} \left( \cos x \times \sin y \right) \] Using the product rule on the right-hand side: \[ \frac{dy}{dx} = \left( -\sin x \times \sin y \right) + \left( \cos x \times \cos y \frac{dy}{dx} \right) \] Now, solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} - \cos x \times \cos y \frac{dy}{dx} = -\sin x \times \sin y \] \[ \frac{dy}{dx} \left( 1 - \cos x \times \cos y \right) = -\sin x \times \sin y \] Thus, \[ \frac{dy}{dx} = \frac{-\sin x \times \sin y}{1 - \cos x \times \cos y} \] Now, substituting \( x = \frac{\pi}{6} \) and \( y = \frac{\pi}{5} \): \[ \frac{dy}{dx} = \frac{-\sin \frac{\pi}{6} \times \sin \frac{\pi}{5}}{1 - \cos \frac{\pi}{6} \times \cos \frac{\pi}{5}} \] Simplifying: \[ \frac{dy}{dx} = -\frac{\frac{1}{2} \times \sin \frac{\pi}{5}}{1 - \frac{\sqrt{3}}{2} \times \cos \frac{\pi}{5}} \] 
Thus, the value of \( \frac{dy}{dx} \) is \( -\frac{1}{2} \).