To find the derivative of \( \sin(5x) \), we apply the chain rule. 

1. The derivative of \( \sin(u) \) is \( \cos(u) \).
2. Here, \( u = 5x \), so the derivative of \( u \) with respect to \( x \) is \( 5 \).

Using the chain rule:

\[
\frac{d}{dx}[\sin(5x)] = \cos(5x) \cdot 5 = 5\cos(5x)
\]

Thus, the derivative of \( \sin(5x) \) is \( 5\cos(5x) \).

Would you like more details or have any questions? Here are some related questions:

1. What is the integral of \( \sin(5x) \)?
2. How do you find the second derivative of \( \sin(5x) \)?
3. Can you explain the chain rule in more detail?
4. What is the derivative of \( \cos(5x) \)?
5. How does the amplitude affect the derivative of a sine function?

**Tip:** When finding derivatives of composite functions, always identify the outer and inner functions clearly to apply the chain rule correctly.

Discover how to find the derivative of sin(5x) using the chain rule. This explanation includes key concepts from calculus and trigonometric functions, making it ideal for advanced students in Grades 11-12.

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