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.