Question

If \(y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \dots \infty}}}\) then \( \frac{dy}{dx} = \)

• A. \(\frac{\cos x}{1 - 2y}\)
• B. \(\frac{\sin x}{1 - 2y}\)
• C. \(-\frac{\sin x}{1 - 2y}\)
• D. \(-\frac{\cos x}{1 - 2y}\)

Hint:

For any function of the form \(y = \sqrt{f(x) + \sqrt{f(x) + \dots}}\), the derivative follows the pattern \(\frac{dy}{dx} = \frac{f'(x)}{2y-1}\). Memorizing this general form can lead to an instant answer for such problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Find slope of normal to x²/³+y²/³=2 at (1,1).

Step 2: Key Formula (Alternate):
Implicit differentiation gives tangent slope m_T. Normal slope m_N = -1/m_T.

Step 3: Detailed Explanation:
(2/3)x⁻¹/³+(2/3)y⁻¹/³·dy/dx=0 → dy/dx = -(y/x)¹/³. At (1,1): m_T=-1. m_N=-1/(-1)=1.

Step 4: Final Answer:
Slope of normal is 1.