If y={x+{x+{x+∞}}} then \;d y/d x is

Correct Answer is : 12y−11/2 y-12y−11

Correct Answer is : 12y−11/2 y-12y−11

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Differential Calculus

Question : 17 of 69

Marks: +1, -0

y=\sqrt{x+\sqrt{x+\sqrt{x+\dots\infty}}}

\;\frac{d y}{d x}

Solution:

Concept:

Represent the infinite expression as

y = \sqrt{x + y}

Explanation:

The given infinite expression can be written as

y = \sqrt{x + \sqrt{x + \sqrt{x + \dots}}} = \sqrt{x + y}

Squaring both sides:

y^2 = x + y

Rearrange:

y^2 - y = x

Differentiate implicitly with respect to

x

2y \frac{dy}{dx} - \frac{dy}{dx} = 1

Factor:

\frac{dy}{dx}(2y - 1) = 1

Solve:

\frac{dy}{dx} = \frac{1}{2y - 1}

Answer:

\frac{1}{2y-1}

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