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The given function is \( f(x) = 5 \cos(7 \ln(x)) \), and you're asked to find \( f'(x) \), the derivative of \( f(x) \).

### Solution:

1. **Differentiate the constant multiplier**:  
   The constant \( 5 \) can be factored out when differentiating:
   \[
   f'(x) = 5 \cdot \frac{d}{dx} [\cos(7 \ln(x))]
   \]

2. **Differentiate the cosine function**:  
   Using the chain rule, the derivative of \( \cos(u) \) is \( -\sin(u) \) times the derivative of \( u \). Here, \( u = 7 \ln(x) \), so:
   \[
   \frac{d}{dx}[\cos(7 \ln(x))] = -\sin(7 \ln(x)) \cdot \frac{d}{dx}[7 \ln(x)]
   \]

3. **Differentiate the natural logarithm**:  
   The derivative of \( 7 \ln(x) \) is \( \frac{7}{x} \), so:
   \[
   \frac{d}{dx}[7 \ln(x)] = \frac{7}{x}
   \]

4. **Final expression**:  
   Now, substitute the result back into the derivative expression:
   \[
   f'(x) = 5 \cdot \left(-\sin(7 \ln(x)) \cdot \frac{7}{x}\right)
   \]
   Simplifying:
   \[
   f'(x) = -\frac{35 \sin(7 \ln(x))}{x}
   \]

### Final Answer:
\[
f'(x) = -\frac{35 \sin(7 \ln(x))}{x}
\]

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:
1. How do you apply the chain rule to composite functions?
2. What is the derivative of other trigonometric functions involving logarithms?
3. How would you find the second derivative of \( f(x) = 5 \cos(7 \ln(x)) \)?
4. What are the applications of differentiating functions involving logarithms?
5. How does the product rule differ from the chain rule in derivatives?

**Tip**: When differentiating composite functions, always identify the inner and outer functions to apply the chain rule correctly.

Find the derivative of the function f(x) = 5 cos(7 ln(x)) using the chain rule and logarithmic function differentiation. Step-by-step solution provided, ideal for students in advanced high school or college calculus.

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