Defreantiation of combosit function cos5x

To differentiate the composite function \( \cos(5x) \), we again apply the chain rule.

Given \( f(x) = \cos(5x) \), where \( g(x) = 5x \), the chain rule tells us:

\[
\frac{d}{dx} [\cos(g(x))] = -\sin(g(x)) \cdot g'(x)
\]

For the given function, \( g(x) = 5x \), so:

\[
g'(x) = \frac{d}{dx} [5x] = 5
\]

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

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

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

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. How do you differentiate \( \cos(2x) \)?
2. What is the derivative of \( \sin(5x) \)?
3. How do you apply the chain rule to functions like \( \cos(ax + b) \)?
4. What is the derivative of \( \cos^{-1}(x) \)?
5. How does the product rule work when combined with trigonometric functions?
6. What is the second derivative of \( \cos(5x) \)?
7. Can you derive the derivative using trigonometric identities?
8. How would you differentiate \( \cos^2(5x) \)?

**Tip:** When using the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.

Learn how to differentiate the composite function cos(5x) using the chain rule. Step-by-step guide with detailed explanations and examples. Suitable for advanced high school students and anyone studying calculus and trigonometry.

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