Given \( y = \sin(2x) \), to find the derivative \( \frac{dy}{dx} \), we apply the chain rule.

The chain rule states that if \( y = f(g(x)) \), then:

\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
\]

Here, \( f(u) = \sin(u) \) and \( g(x) = 2x \), so applying the chain rule:

1. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), so \( f'(g(x)) = \cos(2x) \).
2. The derivative of \( 2x \) with respect to \( x \) is 2.

Therefore, combining these:

\[
\frac{dy}{dx} = \cos(2x) \cdot 2 = 2\cos(2x)
\]

So, \( \frac{dy}{dx} = 2\cos(2x) \).

Would you like a more detailed explanation or any clarifications?

### 5 Related Questions:
1. How do you differentiate \( y = \cos(3x) \)?
2. What is the second derivative of \( y = \sin(2x) \)?
3. How does the chain rule apply in multi-variable calculus?
4. What is the derivative of \( y = \sin^2(x) \)?
5. Can you explain how to find the derivative of trigonometric functions?

### Tip:
Always remember to check whether you need to apply the chain rule when differentiating composite functions.

Discover how to differentiate the function y = sin(2x) using the chain rule in calculus. This step-by-step guide explains key trigonometric functions, the chain rule, and provides the full solution to find dy/dx = 2cos(2x). Perfect for high school and college calculus students.

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