The derivative of `f(x) = "sin" 2x` is

Options

A. cos 2X
B. 2 cos 2X
C. tan 2X
D. `("cos"2x)/(2)`

Answer

To find the derivative of the function \( f(x) = \sin(2x) \), we will use the chain rule. Here are the steps to solve the problem:

### Step-by-Step Solution:

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

2. **Differentiate using the chain rule**: The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative \( f'(g(x)) \cdot g'(x) \).

- Here, let \( g(x) = 2x \).
- The outer function is \( \sin(u) \) where \( u = g(x) = 2x \).

3. **Differentiate the outer function**: The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). Therefore,
\[ \frac{d}{du}(\sin(2x)) = \cos(2x). \]

4. **Differentiate the inner function**: Now we need to differentiate \( g(x) = 2x \). The derivative of \( 2x \) is:
\[ g'(x) = 2. \]

5. **Apply the chain rule**: Now we can combine these results using the chain rule:
\[ f'(x) = \frac{d}{du}(\sin(2x)) \cdot g'(x) = \cos(2x) \cdot 2. \]

6. **Final result**: Therefore, the derivative of \( f(x) = \sin(2x) \) is:
\[ f'(x) = 2 \cos(2x). \]

### Final Answer:
The derivative of \( f(x) = \sin(2x) \) is \( f'(x) = 2 \cos(2x) \).

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