If `y= sqrt"" ( sin x+ y) , then
dy //dx =`

Options

A. `sin x // ( 2y -1)`
B. ` sin x // (1 -2y)`
C. ` cos x// (1-2y)`
D. ` cos x// (2y-1)`

Answer

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

### Step 1: Square both sides
We start with the equation:
\[ y = \sqrt{\sin x + y} \]
To eliminate the square root, we square both sides:
\[ y^2 = \sin x + y \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate all terms on one side:
\[ y^2 - y - \sin x = 0 \]

### Step 3: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides of the equation with respect to \( x \). We will use implicit differentiation:
\[ \frac{d}{dx}(y^2) - \frac{d}{dx}(y) - \frac{d}{dx}(\sin x) = 0 \]
Using the chain rule, we get:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} - \cos x = 0 \]

### Step 4: Factor out \( \frac{dy}{dx} \)
Now, we can factor out \( \frac{dy}{dx} \):
\[ (2y - 1) \frac{dy}{dx} - \cos x = 0 \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives us:
\[ (2y - 1) \frac{dy}{dx} = \cos x \]
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 1} \]

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

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