The expression you provided is:

\[
e^x - 4 \ln(\cos(x))
\]

To analyze or differentiate it further, we can break it down into two main parts:

1. \( e^x \), the exponential function.
2. \( -4 \ln(\cos(x)) \), where \( \ln \) is the natural logarithm of \( \cos(x) \).

Would you like me to simplify, differentiate, or provide more details about this expression?

### Related Questions:
1. How do we differentiate the exponential function \( e^x \)?
2. What is the derivative of \( \ln(\cos(x)) \)?
3. How can we apply the chain rule in differentiation?
4. How do logarithmic and trigonometric functions interact in this context?
5. Can this expression be integrated? If so, what is the result?

**Tip:** Always apply the chain rule carefully when differentiating compositions of functions like \( \ln(\cos(x)) \).

Learn how to differentiate the expression e^x - 4 ln(cos(x)). This solution explores key calculus concepts like exponential, logarithmic, and trigonometric functions, focusing on applying the chain rule and logarithmic differentiation. Suitable for high school and introductory calculus students.

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