If `siny+2x=e^x` then find `dy/dx`

Answer

To find \( \frac{dy}{dx} \) for the equation \( \sin y + 2x = e^x \), we will differentiate both sides with respect to \( x \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sin y + 2x = e^x \]

2. **Differentiate both sides with respect to \( x \)**:
- The left-hand side involves \( \sin y \) and \( 2x \):
- The derivative of \( \sin y \) with respect to \( x \) is \( \cos y \cdot \frac{dy}{dx} \) (using the chain rule).
- The derivative of \( 2x \) is \( 2 \).
- The right-hand side is \( e^x \), whose derivative is \( e^x \).

Therefore, differentiating both sides gives:
\[ \cos y \cdot \frac{dy}{dx} + 2 = e^x \]

3. **Rearrange the equation to isolate \( \frac{dy}{dx} \)**:
\[ \cos y \cdot \frac{dy}{dx} = e^x - 2 \]

4. **Solve for \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{e^x - 2}{\cos y} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{e^x - 2}{\cos y} \]