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Let's solve this problem by differentiating the function \( f(x) = \sin(5 \ln(x)) \) with respect to \( x \).

### Solution

1. **Identify the outer and inner functions**:
   - The outer function is \( \sin(u) \) where \( u = 5 \ln(x) \).
   - The inner function is \( u = 5 \ln(x) \).

2. **Apply the chain rule**:
   According to the chain rule:
   \[
   f'(x) = \cos(5 \ln(x)) \cdot \frac{d}{dx}[5 \ln(x)]
   \]

3. **Differentiate the inner function** \( 5 \ln(x) \):
   \[
   \frac{d}{dx}[5 \ln(x)] = 5 \cdot \frac{1}{x} = \frac{5}{x}
   \]

4. **Combine the results**:
   Substitute back into the derivative:
   \[
   f'(x) = \cos(5 \ln(x)) \cdot \frac{5}{x}
   \]

5. **Final Answer**:
   \[
   f'(x) = \frac{5 \cos(5 \ln(x))}{x}
   \]

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So, the derivative of \( f(x) = \sin(5 \ln(x)) \) is:
\[
f'(x) = \frac{5 \cos(5 \ln(x))}{x}
\]

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Would you like further details on the steps, or do you have any additional questions? 

Here are some related questions to practice:

1. Differentiate \( g(x) = \cos(3 \ln(x)) \).
2. Differentiate \( h(x) = e^{2 \ln(x)} \).
3. Find the second derivative of \( f(x) = \sin(5 \ln(x)) \).
4. Differentiate \( k(x) = \tan(4 \ln(x)) \).
5. Solve \( f'(x) = 0 \) for \( f(x) = \sin(5 \ln(x)) \).

**Tip:** For functions involving \( \ln(x) \) inside trigonometric or exponential functions, the chain rule is essential to handle the differentiation.

Differentiate the function f(x) = sin(5 ln(x)).

Learn how to differentiate the function f(x) = sin(5 ln(x)) using the chain rule. This solution covers key calculus concepts, including trigonometric and logarithmic functions, and applies the chain rule for differentiation. Suitable for students in Grades 11-12.

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