Find the derivatives of the following :
`1/2sin2x`

Answer

To find the derivative of the function \( y = \frac{1}{2} \sin(2x) \), we will use the chain rule and the fact that the derivative of \( \sin(x) \) is \( \cos(x) \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = \frac{1}{2} \sin(2x) \]

2. **Differentiate with respect to \( x \)**:
We apply the derivative to \( y \):
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{d}{dx}[\sin(2x)] \]

3. **Use the chain rule**:
The derivative of \( \sin(u) \) is \( \cos(u) \cdot \frac{du}{dx} \), where \( u = 2x \). Thus, we have:
\[ \frac{d}{dx}[\sin(2x)] = \cos(2x) \cdot \frac{d}{dx}(2x) \]
Here, \( \frac{d}{dx}(2x) = 2 \).

4. **Combine the results**:
Substituting back, we get:
\[ \frac{d}{dx}[\sin(2x)] = \cos(2x) \cdot 2 \]
Therefore,
\[ \frac{dy}{dx} = \frac{1}{2} \cdot (2 \cos(2x)) \]

5. **Simplify the expression**:
The \( \frac{1}{2} \) and \( 2 \) cancel out:
\[ \frac{dy}{dx} = \cos(2x) \]

### Final Answer:
The derivative of \( y = \frac{1}{2} \sin(2x) \) is:
\[ \frac{dy}{dx} = \cos(2x) \]