GATE AG 2010 • Q29 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Limit, continuity and differentiability

Question Statement:

The derivative of $y = \sqrt{x + \sqrt{x + \sqrt{x + \dots}}}$ with respect to $x$ at $y = 0$ is:

Options:

  • (A) −1
  • (B) 0
  • (C) 1
  • (D) 2

Official Answer Key:

A

Step-by-Step Solution & Explanation

The infinite nested radical satisfies y² = x + y. Differentiating implicitly: 2y·y′ = 1 + y′ ⇒ y′(2y−1) = 1 ⇒ y′ = 1/(2y−1). At y = 0: y′ = 1/(0−1) = −1.

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