Find `dy/dx if y=8sin2x`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = 8 \sin(2x) \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the function**: We have \( y = 8 \sin(2x) \).

2. **Apply the constant multiple rule**: Since 8 is a constant, we can factor it out when differentiating. Thus, we can express the derivative as:
\[ \frac{dy}{dx} = 8 \cdot \frac{d}{dx}(\sin(2x)) \]

3. **Differentiate the sine function**: The derivative of \( \sin(\theta) \) with respect to \( \theta \) is \( \cos(\theta) \). Here, \( \theta = 2x \). Therefore:
\[ \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot \frac{d}{dx}(2x) \]

4. **Differentiate the angle**: The derivative of \( 2x \) with respect to \( x \) is 2. Thus, we have:
\[ \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot 2 \]

5. **Combine the results**: Now substituting back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 8 \cdot (2 \cos(2x)) = 16 \cos(2x) \]

6. **Final result**: Therefore, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 16 \cos(2x) \]