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If y = sqrt(x + sqrt(y + sqrt(x + sqrt(y +... to infinity)))), then dy/dx equals:

1. 1/(2y - 1)
2. (y² - x) / (2y³ - 2xy - 1)
3. (2y - 1)
4. None of these

Correct answer: None of these

Solution

From self-similarity: y² = x + sqrt(y + y²). Let y² - x = sqrt(y + y²), so (y² - x)² = y + y². Differentiating implicitly: 2(y² - x)(2y*y' - 1) = y'(1 + 2y). Solving: y'[4y(y² - x) - (1 + 2y)] = 2(y² - x), so y' = 2(y² - x) / [4y³ - 4xy - 2y - 1]. This does not match any of options A, B, or C exactly.

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