If `y=sqrt(x+sqrt(x+sqrt(+…..oo)))` then prove that `(2y-1)dy/dx=1.`

Explanation

To prove that \( (2y - 1) \frac{dy}{dx} = 1 \) given that \( y = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} \), we can follow these steps:

### Step 1: Define the equation
We start with the definition of \( y \):
\[ y = \sqrt{x + y} \]
This is because the nested square roots converge to \( y \).
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