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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.

Differentiate the function: f(x) = ln(cos(x))

Learn how to differentiate the function f(x) = ln(cos(x)) with a step-by-step solution using the chain rule. This calculus problem involves key concepts such as differentiation and trigonometric functions, suitable for students in Grades 11-12.

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