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To find the derivative of the function
\[
f(x) = \cos^{-1}\left(e^{3 \cos x}\right),
\]
we'll use the chain rule and the derivative formulas for the inverse cosine function and the exponential function.

### Step-by-Step Solution
1. **Identify the outer function and the inner function**:
   - The outer function is \( g(u) = \cos^{-1}(u) \).
   - The inner function is \( u = e^{3 \cos x} \).

2. **Derivative of the Outer Function** \( g(u) = \cos^{-1}(u) \):
   - The derivative of \( \cos^{-1}(u) \) with respect to \( u \) is
     \[
     g'(u) = -\frac{1}{\sqrt{1 - u^2}}.
     \]

3. **Derivative of the Inner Function** \( u = e^{3 \cos x} \):
   - To find \( u'(x) \), we need to apply the chain rule again because \( u = e^{3 \cos x} \) is itself a composition of functions.
   - Let \( v = 3 \cos x \). Then \( u = e^v \).
   - The derivative of \( e^v \) with respect to \( x \) is:
     \[
     \frac{d}{dx} e^{3 \cos x} = e^{3 \cos x} \cdot \frac{d}{dx}(3 \cos x).
     \]
   - Now, \( \frac{d}{dx}(3 \cos x) = -3 \sin x \).
   - So,
     \[
     u'(x) = e^{3 \cos x} \cdot (-3 \sin x) = -3 e^{3 \cos x} \sin x.
     \]

4. **Applying the Chain Rule**:
   - By the chain rule, \( f'(x) = g'(u) \cdot u'(x) \).
   - Substitute \( g'(u) = -\frac{1}{\sqrt{1 - u^2}} \) and \( u'(x) = -3 e^{3 \cos x} \sin x \):
     \[
     f'(x) = -\frac{1}{\sqrt{1 - \left(e^{3 \cos x}\right)^2}} \cdot \left(-3 e^{3 \cos x} \sin x\right).
     \]
   - Simplifying, we get:
     \[
     f'(x) = \frac{3 e^{3 \cos x} \sin x}{\sqrt{1 - \left(e^{3 \cos x}\right)^2}}.
     \]

### Final Answer
\[
f'(x) = \frac{3 e^{3 \cos x} \sin x}{\sqrt{1 - e^{6 \cos x}}}.
\]

Would you like further details on any part of this solution?

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Here are 5 related questions for further exploration:
1. What is the chain rule, and how is it applied in finding derivatives?
2. How do we handle derivatives of functions that involve inverse trigonometric functions?
3. Can we apply a similar process to find the derivative of \( f(x) = \sin^{-1}(e^{3 \sin x}) \)?
4. How does the exponential function affect the differentiation process?
5. What other types of functions require the chain rule for differentiation?

**Tip:** For composite functions, always identify the outer and inner functions clearly before applying the chain rule.

Evaluate the derivative of the function f(x) = cos^(-1)(e^(3 cos x)).

Learn how to find the derivative of f(x) = cos^(-1)(e^(3 cos x)) using the chain rule and formulas for inverse trigonometric and exponential functions. This solution provides a detailed breakdown of each step, making it ideal for students in college-level calculus.

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