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

Options

A. `( sin^2 ( a +y))/(sin a)`
B. ` sin (a +y)`
C. `sin^2 (a+y)`
D. ` (sin (a+ y))/( sin a)`

Answer

To solve the equation \( \sin y = x \sin(a + y) \) and find \( \frac{dy}{dx} \), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides with respect to \( x \)

Starting with the equation:
\[ \sin y = x \sin(a + y) \]

We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\sin y) = \frac{d}{dx}(x \sin(a + y)) \]

### Step 2: Apply the chain rule on the left side

Using the chain rule on the left side:
\[ \cos y \frac{dy}{dx} = \frac{d}{dx}(x \sin(a + y)) \]

### Step 3: Differentiate the right side using the product rule

For the right side, we apply the product rule:
\[ \frac{d}{dx}(x \sin(a + y)) = \sin(a + y) + x \cos(a + y) \frac{dy}{dx} \]

### Step 4: Set the derivatives equal

Now we have:
\[ \cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx} \]

### Step 5: Rearrange the equation to isolate \( \frac{dy}{dx} \)

Rearranging gives:
\[ \cos y \frac{dy}{dx} - x \cos(a + y) \frac{dy}{dx} = \sin(a + y) \]

Factoring out \( \frac{dy}{dx} \):
\[ \left( \cos y - x \cos(a + y) \right) \frac{dy}{dx} = \sin(a + y) \]

### Step 6: Solve for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)} \]

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

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