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

Answer

To find \( \frac{dy}{dx} \) for the function \( y = x - \cos(2x) \), we will differentiate the function step by step.

### Step-by-Step Solution:

1. **Write down the function**:
We start with the given function:
\[ y = x - \cos(2x) \]

2. **Differentiate each term**:
We will differentiate \( y \) with respect to \( x \). This means we will differentiate \( x \) and \( -\cos(2x) \) separately.
- The derivative of \( x \) with respect to \( x \) is \( 1 \).
- For \( -\cos(2x) \), we will use the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \), where \( u = 2x \). We also need to multiply by the derivative of \( u \) with respect to \( x \), which is \( 2 \).

3. **Apply the differentiation**:
Using the above rules, we differentiate:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}(\cos(2x)) \]
\[ \frac{dy}{dx} = 1 - (-\sin(2x) \cdot 2) \]

4. **Simplify the expression**:
Now simplify the expression:
\[ \frac{dy}{dx} = 1 + 2\sin(2x) \]

5. **Final result**:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 1 + 2\sin(2x) \]