Find `dy/dx if 2y= x-cos2x`

Answer

To find \(\frac{dy}{dx}\) given the equation \(2y = x - \cos(2x)\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ 2y = x - \cos(2x) \]

2. **Differentiate both sides with respect to \(x\)**:
- The left side: Using the chain rule, the derivative of \(2y\) with respect to \(x\) is:
\[ \frac{d}{dx}(2y) = 2 \frac{dy}{dx} \]
- The right side: Differentiate \(x\) and \(-\cos(2x)\):
- The derivative of \(x\) is \(1\).
- For \(-\cos(2x)\), we apply the chain rule:
\[ \frac{d}{dx}(-\cos(2x)) = -(-\sin(2x)) \cdot \frac{d}{dx}(2x) = \sin(2x) \cdot 2 = 2\sin(2x) \]
Thus, the derivative of the right side is:
\[ 1 + 2\sin(2x) \]

3. **Set the derivatives equal**:
\[ 2 \frac{dy}{dx} = 1 + 2\sin(2x) \]

4. **Solve for \(\frac{dy}{dx}\)**:
Divide both sides by \(2\):
\[ \frac{dy}{dx} = \frac{1 + 2\sin(2x)}{2} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{1}{2} + \sin(2x) \]