To find the derivative of \( f(x) = \ln |\cos(\ln x)| \), we will use the chain rule and the derivative of the natural logarithm function.

### Steps:

1. **Outer function**: The derivative of \( \ln |u(x)| \) is \( \frac{1}{u(x)} \cdot \frac{du}{dx} \), where \( u(x) = \cos(\ln x) \).
   
   So, we get:
   \[
   \frac{d}{dx} \ln |\cos(\ln x)| = \frac{1}{\cos(\ln x)} \cdot \frac{d}{dx} [\cos(\ln x)].
   \]

2. **Derivative of \( \cos(\ln x) \)**: Use the chain rule again. The derivative of \( \cos u \) is \( -\sin u \), and the derivative of \( \ln x \) is \( \frac{1}{x} \).
   
   Thus, \( \frac{d}{dx} \cos(\ln x) = -\sin(\ln x) \cdot \frac{1}{x} \).

3. **Combine the results**:
   \[
   \frac{d}{dx} \ln |\cos(\ln x)| = \frac{-\sin(\ln x)}{x \cos(\ln x)}.
   \]

Thus, the derivative of \( \ln |\cos(\ln x)| \) is:
\[
f'(x) = \frac{-\sin(\ln x)}{x \cos(\ln x)}.
\]

### Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you apply the chain rule to composite functions in other examples?
2. What is the derivative of \( \ln |\sin(\ln x)| \)?
3. How do you differentiate a product or quotient of logarithmic and trigonometric functions?
4. What are the properties of the natural logarithm function that make it useful in calculus?
5. How does the absolute value affect the differentiation of a function?

**Tip:** The chain rule is essential for differentiating composite functions. Always identify the outer and inner functions before applying it.

Learn how to differentiate ln|cos(ln x)| using the chain rule, logarithmic differentiation, and trigonometric derivatives. This solution breaks down each step of finding the derivative, combining key calculus principles. Suitable for advanced high school and college-level students.

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