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To differentiate \( f(x) = \ln \left[ x^8 (x + 2)^6 (x^2 + 6)^5 \right] \), we can use the chain rule along with the properties of logarithms to simplify our work.

### Step-by-Step Solution

1. **Simplify Using Logarithmic Properties:**
   Rewrite \( f(x) \) by expanding the logarithm:
   \[
   f(x) = \ln(x^8) + \ln((x + 2)^6) + \ln((x^2 + 6)^5)
   \]
   Using the power rule of logarithms \( \ln(a^b) = b \ln(a) \), we get:
   \[
   f(x) = 8 \ln(x) + 6 \ln(x + 2) + 5 \ln(x^2 + 6)
   \]

2. **Differentiate Term by Term:**
   Now, differentiate each term individually.

   - For \( 8 \ln(x) \):
     \[
     \frac{d}{dx} [8 \ln(x)] = \frac{8}{x}
     \]

   - For \( 6 \ln(x + 2) \):
     \[
     \frac{d}{dx} [6 \ln(x + 2)] = \frac{6}{x + 2}
     \]

   - For \( 5 \ln(x^2 + 6) \):
     \[
     \frac{d}{dx} [5 \ln(x^2 + 6)] = 5 \cdot \frac{2x}{x^2 + 6} = \frac{10x}{x^2 + 6}
     \]

3. **Combine the Results:**
   Now, sum up all the differentiated terms:
   \[
   f'(x) = \frac{8}{x} + \frac{6}{x + 2} + \frac{10x}{x^2 + 6}
   \]

### Final Answer
\[
f'(x) = \frac{8}{x} + \frac{6}{x + 2} + \frac{10x}{x^2 + 6}
\]

Would you like a detailed explanation of each step, or do you have any further questions?

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Here are some related questions you might consider:
1. How would we differentiate if \( f(x) \) had more complex exponents on each term?
2. What if the logarithmic function was a natural logarithm of a quotient instead of a product?
3. Can this method be applied if \( f(x) \) had more variables, not just \( x \)?
4. How would this process change if we were using base 10 logarithms instead of natural logarithms?
5. What if we wanted to integrate \( f(x) \) instead of differentiating it?

**Tip:** Using logarithmic differentiation can greatly simplify differentiating complex products or quotients by reducing them to simpler additive or subtractive forms.

Let \( f(x) = \ln \left[ x^8 (x + 2)^6 (x^2 + 6)^5 \right] \). Find \( f'(x) \).

Learn how to find the derivative of \( f(x) = \ln \left[ x^8 (x + 2)^6 (x^2 + 6)^5 \right] \) by applying logarithmic differentiation. This solution includes using logarithmic properties, the chain rule, and simplifies each term for a step-by-step understanding. Suitable for calculus students.

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