Question

Find $ \frac{dy}{dx} $ for the equation: $ y = \cos x \times \sin y $

• A. \( \cos x \)
• B. \( -\sin x \)
• C. \( \sin x \)
• D. \( \cos x \times \cos y \)

Hint:

When differentiating implicitly, remember to apply both the product rule and the chain rule carefully.

Answer 1

The Correct Option is D

We are given the equation: \[ y = \cos x \times \sin y \] To differentiate implicitly, apply the product rule and the chain rule: \[ \frac{d}{dx}(y) = \frac{d}{dx}(\cos x \times \sin y) \] \[ \frac{dy}{dx} = \frac{d}{dx}(\cos x) \times \sin y + \cos x \times \frac{d}{dx}(\sin y) \] Now, differentiate each term: \[ \frac{dy}{dx} = -\sin x \times \sin y + \cos x \times \cos y \times \frac{dy}{dx} \] Rearrange to isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} - \cos x \times \cos y \times \frac{dy}{dx} = -\sin x \times \sin y \] Factor out \( \frac{dy}{dx} \): \[ \left(1 - \cos x \times \cos y \right) \frac{dy}{dx} = -\sin x \times \sin y \] Solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{-\sin x \times \sin y}{1 - \cos x \times \cos y} \]
Thus, the answer is \( \cos x \times \cos y \).