Differentiate the following functions w.r.t. x:
`sin 2x`

Answer

To differentiate the function \( y = \sin(2x) \) with respect to \( x \), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = \sin(2x) \).

2. **Differentiate using the chain rule**:
The chain rule states that if you have a composite function \( y = \sin(u) \) where \( u = 2x \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \cos(u) \cdot \frac{du}{dx} \]
Here, \( u = 2x \).

3. **Differentiate \( u = 2x \)**:
The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = 2 \]

4. **Substitute back into the derivative**:
Now substituting \( u = 2x \) back into the derivative:
\[ \frac{dy}{dx} = \cos(2x) \cdot 2 \]

5. **Simplify the expression**:
Therefore, we have:
\[ \frac{dy}{dx} = 2 \cos(2x) \]

### Final Answer:
Thus, the derivative of \( \sin(2x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2 \cos(2x) \]