If `y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+...oo))))`, then `(dy)/(dx)` is equal to

Options

A. `(y+x)/(y^(2)-2)`
B. `(y^(3)-x)/(2y^(2)-2xy-1)`
C. `(y^(3)+x)/(2y^(2)-x)`
D. none of these

Answer

To solve the problem where \( y = \sqrt{x + \sqrt{y + \sqrt{x + \sqrt{y + \ldots}}}} \), we can start by simplifying the expression.

### Step 1: Rewrite the equation
We can denote the infinite nested square root as follows:
\[ y = \sqrt{x + y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = x + y \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ y^2 - y - x = 0 \]

### Step 4: Differentiate implicitly
Now, we will differentiate both sides of the equation with respect to \( x \). Using implicit differentiation:
\[ \frac{d}{dx}(y^2) - \frac{d}{dx}(y) - \frac{d}{dx}(x) = 0 \]
Applying the chain rule:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} - 1 = 0 \]

### Step 5: Factor out \(\frac{dy}{dx}\)
Now, we can factor out \(\frac{dy}{dx}\):
\[ (2y - 1) \frac{dy}{dx} - 1 = 0 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Rearranging gives:
\[ (2y - 1) \frac{dy}{dx} = 1 \]
Thus,
\[ \frac{dy}{dx} = \frac{1}{2y - 1} \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{2y - 1} \]