If y = √ x + √ y + √ x + √ y + . . . . ∞ , then d y d x is equal to Options: A . &nbsp12y−1 B . &nbspy2−x2y3−2xy−1 C . &nbsp(2y−1) D . &nbspNone of these

If y=√x+√y+√x+√y+....∞, then dydx is equal to

Question

y = \sqrt{x + \sqrt{y + \sqrt{x + \sqrt{y + . . . . \infty,}}}}

then

\frac{d y}{d x}

is equal to

Options:

A . 12y−1

B . y2−x2y3−2xy−1

C . (2y−1)

D . None of these

Answer: Option B
:
B

∴ y = \sqrt{x + \sqrt{y + y}}

\Rightarrow (y^{2} - x) = \sqrt{2 y}

o r (y^{2} - x)^{2} = 2 y

Differentiating both sides w.r.t. x, then

2 (y^{2} - x) (2 y \frac{d y}{d x} - 1) = 2 \frac{d y}{d x}

∴ \frac{d y}{d x} = \frac{(y^{2} - x)}{2 y^{3} - 2 x y - 1}

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