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

Options

A. `(cosx)/(2y-1)`
B. `(cosx)/(1-2y)`
C. `(sinx)/(1-2y)`
D. `(sinx)/(2y-1)`

Explanation

To find \(\frac{dy}{dx}\) for the equation \(y = \sqrt{\sin x + y}\), we will follow these steps:

### Step 1: Square both sides
Start with the equation:
\[ y = \sqrt{\sin x + y} \]
Square both sides to eliminate the square root:
\[ y^2 = \sin x + y \]

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

### Step 3: Differentiate both sides
Now, differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(y^2) - \frac{d}{dx}(y) - \frac{d}{dx}(\sin x) = 0 \]
Using the chain rule, we have:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} - \cos x = 0 \]

### Step 4: Factor out \(\frac{dy}{dx}\)
Rearranging gives:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} = \cos x \]
Factoring out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}(2y - 1) = \cos x \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Now, isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 1} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 1} \]