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We are tasked with finding the derivative \( f'(x) \) for the function \( f(x) = \ln\left((x^2 + 4)^3\right) \).

### Steps to solve:
1. **Simplify the logarithmic function**:  
   Using the property of logarithms, \( \ln(a^b) = b \ln(a) \), we can simplify:
   \[
   f(x) = \ln\left((x^2 + 4)^3\right) = 3 \ln(x^2 + 4)
   \]

2. **Differentiate** \( f(x) = 3 \ln(x^2 + 4) \):
   To find the derivative, we apply the chain rule.
   \[
   f'(x) = 3 \cdot \frac{d}{dx} \left[\ln(x^2 + 4)\right]
   \]
   The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = x^2 + 4 \).

   First, calculate \( \frac{d}{dx}(x^2 + 4) \):
   \[
   \frac{d}{dx}(x^2 + 4) = 2x
   \]

   Now, apply this to the chain rule:
   \[
   f'(x) = 3 \cdot \frac{1}{x^2 + 4} \cdot 2x = \frac{6x}{x^2 + 4}
   \]

Thus, the derivative is:
\[
f'(x) = \frac{6x}{x^2 + 4}
\]

Let me know if you need more details, or if you'd like to explore further!

Here are 5 related questions to practice:
1. How would the derivative change if the power inside the logarithm was \( 5 \) instead of \( 3 \)?
2. What is the second derivative \( f''(x) \) of this function?
3. Can you generalize the result for \( f(x) = \ln\left((x^2 + c)^n\right) \) for any constants \( c \) and \( n \)?
4. What happens if you differentiate \( f(x) = \ln\left((x^n + b)^k\right) \)?
5. How would the derivative change if the function was \( \ln(x^2 + 4) \) without the exponent \( 3 \)?

**Tip**: Always simplify logarithmic expressions when possible before differentiating to make the process easier.

Find the derivative f'(x) for f(x) = ln((x^2 + 4)^3).

Learn how to find the derivative of f(x) = ln((x^2 + 4)^3) using logarithmic properties and the chain rule. This calculus problem explores key concepts like the logarithmic differentiation and the chain rule.

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