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MHT CET202523 Apr 2025Morning ShiftMathematicsDifferentiationActual

If y = x+ y + x+ y + , then dy d x =

Options

  1. A2 y-1
  2. B1 2 y-1
  3. Cy^2-x 2 y^3-2 x y-1
  4. D¹⁴ C₆

Correct answer

C. y^2-x 2 y^3-2 x y-1

Step-by-step solution

The infinite nested radical y = x + y + x + y + contains a repeating pattern, allowing it to be rewritten as y = x + y + y . This simplifies to y = x + 2y . Squaring both sides yields y^2 = x + 2y . Isolating the radical gives y^2 - x = 2y . Squaring again produces (y^2 - x)^2 = 2y . Differentiating implicitly with respect to x : 2(y^2 - x)(2y dy dx - 1) = 2 dy dx Dividing by 2 and expanding: 2y(y^2 - x) dy dx - (y^2 - x) = dy dx Rearranging terms: dy dx [2y(y^2 - x) - 1] = y^2 - x Solving for the derivative: dy dx

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