If `y=2 +sqrt(sinx+2+sqrt(sinx+2+sqrt(sinx+…oo)))` then the value of `(dy)/(dx)` at x = 0 is

Options

A. 0
B. 2
C. `(1)/(2)`
D. `(1)/(3)`

Answer

To solve the problem, we start with the equation given:

\[ y = 2 + \sqrt{\sin x + 2 + \sqrt{\sin x + 2 + \sqrt{\sin x + \ldots}}} \]

### Step 1: Setting up the equation
We can express the infinite nested square root in terms of \(y\):
\[ y = 2 + \sqrt{\sin x + y} \]

### Step 2: Rearranging the equation
Subtract 2 from both sides:
\[ y - 2 = \sqrt{\sin x + y} \]

### Step 3: Squaring both sides
Square both sides to eliminate the square root:
\[ (y - 2)^2 = \sin x + y \]

### Step 4: Expanding the left side
Expanding the left side gives:
\[ y^2 - 4y + 4 = \sin x + y \]

### Step 5: Rearranging the equation
Rearranging this equation leads to:
\[ y^2 - 5y + 4 - \sin x = 0 \]

### Step 6: Differentiating both sides
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(y^2 - 5y + 4 - \sin x) = 0 \]

Using the chain rule, we get:
\[ 2y \frac{dy}{dx} - 5 \frac{dy}{dx} - \cos x = 0 \]

### Step 7: Factoring out \(\frac{dy}{dx}\)
Factoring out \(\frac{dy}{dx}\):
\[ (2y - 5) \frac{dy}{dx} = \cos x \]

### Step 8: Solving for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 5} \]

### Step 9: Finding \(y\) at \(x = 0\)
To find \(\frac{dy}{dx}\) at \(x = 0\), we need to find the value of \(y\) when \(x = 0\):
\[ y^2 - 5y + 4 - \sin(0) = 0 \]
This simplifies to:
\[ y^2 - 5y + 4 = 0 \]

### Step 10: Solving the quadratic equation
Using the quadratic formula:
\[ y = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2} \]
This gives us:
\[ y = 4 \quad \text{or} \quad y = 1 \]

### Step 11: Choosing the appropriate value for \(y\)
Since the expression for \(y\) involves square roots and must be positive, we take:
\[ y = 4 \]

### Step 12: Substituting \(y\) back into the derivative
Now substituting \(y = 4\) back into the derivative:
\[ \frac{dy}{dx} = \frac{\cos(0)}{2(4) - 5} = \frac{1}{8 - 5} = \frac{1}{3} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = 0\) is:
\[ \frac{1}{3} \]