To differentiate the given function:

$f(x) = \ln(\cos(x))$

we will use the chain rule. The chain rule states that if $f(x) = \ln(g(x))$, then the derivative is:

$f'(x) = \frac{1}{g(x)} \cdot g'(x)$

Step-by-step Solution

1. Let $g(x) = \cos(x)$, so $f(x) = \ln(g(x))$.
2. Differentiate $g(x) = \cos(x)$: $g'(x) = -\sin(x)$
3. Substitute into the chain rule formula: $f'(x) = \frac{1}{\cos(x)} \cdot (-\sin(x))$
4. Simplify the expression: $f'(x) = -\tan(x)$

Answer

The derivative of the function is: $f'(x) = -\tan(x)$

Would you like more details or have any questions about this process?

Here are 5 related questions you might find useful:

1. How do you differentiate other logarithmic functions like $\ln(\sin(x))$?
2. What is the derivative of $\ln(x)$ and how does it compare to this problem?
3. How does the chain rule apply when differentiating nested functions?
4. What are the conditions where the derivative of $\tan(x)$ is undefined?
5. How do you handle the differentiation of functions involving both sine and cosine?

Tip: Remember that using the chain rule correctly often involves finding both the derivative of the outer function and the derivative of the inner function.