Find `dy/dx if y+siny=cosx`

Explanation

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

### Step-by-Step Solution:

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

2. **Differentiate both sides with respect to \(x\):**
- The left side involves \(y\) and \(\sin y\), so we use implicit differentiation.
- The derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx}\).
- The derivative of \(\sin y\) with respect to \(x\) is \(\cos y \cdot \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\cos x\) with respect to \(x\) is \(-\sin x\).

Thus, differentiating both sides gives:
\[ \frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = -\sin x \]

3. **Factor out \(\frac{dy}{dx}\) from the left side:**
\[ \frac{dy}{dx}(1 + \cos y) = -\sin x \]

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

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