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To solve the derivative of the function given:

\[
f(x) = \frac{\ln(x)}{3\ln(x) + 2}
\]

we can use the **quotient rule** for derivatives. The quotient rule states that if \( f(x) = \frac{g(x)}{h(x)} \), then:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]

### Step-by-Step Solution:

1. **Identify \( g(x) \) and \( h(x) \):**
   - \( g(x) = \ln(x) \)
   - \( h(x) = 3\ln(x) + 2 \)

2. **Find \( g'(x) \):**
   - The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
   - So, \( g'(x) = \frac{1}{x} \).

3. **Find \( h'(x) \):**
   - The derivative of \( 3\ln(x) + 2 \) is \( \frac{3}{x} \).
   - So, \( h'(x) = \frac{3}{x} \).

4. **Apply the Quotient Rule:**

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]

Substitute \( g(x) \), \( h(x) \), \( g'(x) \), and \( h'(x) \) into this formula:

\[
f'(x) = \frac{\frac{1}{x}(3\ln(x) + 2) - \ln(x) \cdot \frac{3}{x}}{(3\ln(x) + 2)^2}
\]

5. **Simplify the expression:**

   The numerator becomes:

   \[
   \frac{3\ln(x) + 2 - 3\ln(x)}{x} = \frac{2}{x}
   \]

   So, the derivative simplifies to:

   \[
   f'(x) = \frac{\frac{2}{x}}{(3\ln(x) + 2)^2} = \frac{2}{x(3\ln(x) + 2)^2}
   \]

### Final Answer:

\[
\frac{d}{dx} \left( \frac{\ln(x)}{3 \ln(x) + 2} \right) = \frac{2}{x(3\ln(x) + 2)^2}
\]

Would you like further details or have any questions?

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### Additional Questions to Explore:
1. What is the quotient rule and how is it used in derivatives?
2. How does the derivative of \( \ln(x) \) affect the solution?
3. Can this result be verified by rewriting the function as a product and using the product rule?
4. What would the second derivative of this function look like?
5. How do logarithmic derivatives simplify complex functions?

### Tip:
When differentiating fractions, the quotient rule is especially helpful, but always check if simplifying before differentiation could make the process easier.

Find the derivative of \( \frac{\ln(x)}{3 \ln(x) + 2} \).

Learn how to find the derivative of \( \frac{\ln(x)}{3 \ln(x) + 2} \) with a step-by-step solution using the quotient rule. This solution covers key calculus concepts like logarithmic differentiation and simplifies complex functions. Ideal for advanced high school calculus students.

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