If y-x-y-x-y- thendy - dx is equal toA- 1-2 y-1B- y2-x-2 y3-2 x y-1C- 2 y-1D- none of these

The correct option is B y^2 x/2y^3 2xy 1∴ y = √(x + √(y+y)) ⇒ (y^2 x) = √(2y) or (y^2 x^2) = 2y Differentiating both sides w.r.t. x, then 2 (y^2 1) ...

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Standard XII

Mathematics

Implicit Differentiation

If y=√x+√y+√x...

Question

If $y = \sqrt{x + \sqrt{y + \sqrt{x + \sqrt{y + . . . \infty,}}}}$ then
$\frac{d y}{d x}$ is equal to

\frac{1}{2 y - 1}

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\frac{y^{2} - x}{2 y^{3} - 2 x y - 1}

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(2 y - 1)

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n o n e o f t h e s e

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Solution

The correct option is B $\frac{y^{2} - x}{2 y^{3} - 2 x y - 1}$
$∴ y = \sqrt{x + \sqrt{y + y}} \Rightarrow (y^{2} - x) = \sqrt{2 y}$
or $(y^{2} - x^{2}) = 2 y$
Differentiating both sides w.r.t. x, then
$2 (y^{2} - 1) (2 y \frac{d y}{d x} - 1) = 2 \frac{d y}{d x} ∴ \frac{d y}{d x} = \frac{(y^{2} - x)}{2 y^{3} - 2 x y - 1}$

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