Question

If \( \sin y = x \sin(a + y) \), then \( \frac{dy}{dx} \) is:

• A. \( \frac{\sin(a + y)}{\sin a} \)
• B. \( \frac{\sin^2(a + y)}{\sin a} \)
• C. \( \frac{\cos(a + y)}{\cos a} \)
• D. \( \frac{\cos(a + y)}{\cos a} \)

Hint:

When differentiating implicitly, remember to differentiate both sides of the equation with respect to \( x \) and solve for \( \frac{dy}{dx} \) using the chain rule and product rule.

Answer 1

The Correct Option is C

We are given the equation: \[ \sin y = x \sin(a + y) \] To find \( \frac{dy}{dx} \), we differentiate both sides with respect to \( x \) using implicit differentiation. 1. Differentiating the left-hand side: \[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \] 2. Differentiating the right-hand side using the product rule: \[ \frac{d}{dx}(x \sin(a + y)) = \sin(a + y) + x \cdot \cos(a + y) \cdot \frac{dy}{dx} \] Now, equating both sides: \[ \cos y \cdot \frac{dy}{dx} = \sin(a + y) + x \cdot \cos(a + y) \cdot \frac{dy}{dx} \] 3. Solving for \( \frac{dy}{dx} \): \[ \cos y \cdot \frac{dy}{dx} - x \cdot \cos(a + y) \cdot \frac{dy}{dx} = \sin(a + y) \] Factor out \( \frac{dy}{dx} \): \[ \left( \cos y - x \cdot \cos(a + y) \right) \cdot \frac{dy}{dx} = \sin(a + y) \] Thus, we get: \[ \frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cdot \cos(a + y)} \] Therefore, the correct answer is option (C) \( \frac{\cos(a + y)}{\cos a} \).