Find `dy/dx if y= x-siny`

Answer

To find \( \frac{dy}{dx} \) for the equation \( y = x - \sin y \), we will differentiate both sides with respect to \( x \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ y = x - \sin y \]

2. **Differentiate both sides with respect to \( x \):**
- The left side becomes \( \frac{dy}{dx} \).
- The right side requires the use of the chain rule for \( \sin y \):
\[ \frac{d}{dx}(x) - \frac{d}{dx}(\sin y) = 1 - \cos y \cdot \frac{dy}{dx} \]

Thus, we have:
\[ \frac{dy}{dx} = 1 - \cos y \cdot \frac{dy}{dx} \]

3. **Rearrange the equation to isolate \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = 1 \]
Factor out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(1 + \cos y) = 1 \]

4. **Solve for \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} = \frac{1}{1 + \cos y} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{1}{1 + \cos y} \]