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To solve this derivative problem, let's go through it step-by-step.

### Problem
Find the derivative with respect to \( x \) of the function:
\[
f(x) = \frac{\ln(x)}{\ln(x) + 3}
\]

### Solution
To find \( \frac{d}{dx} \left( \frac{\ln(x)}{\ln(x) + 3} \right) \), we'll use the **quotient rule** for differentiation, which states:
\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}
\]
where:
- \( u = \ln(x) \)
- \( v = \ln(x) + 3 \)

#### Step 1: Differentiate \( u \) and \( v \)
1. **Differentiate \( u = \ln(x) \):**
   \[
   u' = \frac{1}{x}
   \]

2. **Differentiate \( v = \ln(x) + 3 \):**
   \[
   v' = \frac{1}{x}
   \]

#### Step 2: Apply the Quotient Rule
Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule:
\[
f'(x) = \frac{(\ln(x) + 3) \cdot \frac{1}{x} - \ln(x) \cdot \frac{1}{x}}{(\ln(x) + 3)^2}
\]

#### Step 3: Simplify the Expression
Combine terms in the numerator:
\[
f'(x) = \frac{\frac{\ln(x) + 3 - \ln(x)}{x}}{(\ln(x) + 3)^2}
\]
This simplifies further to:
\[
f'(x) = \frac{\frac{3}{x}}{(\ln(x) + 3)^2}
\]
So, the derivative is:
\[
f'(x) = \frac{3}{x (\ln(x) + 3)^2}
\]

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

Would you like further explanation on any part of this solution?

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Here are 5 related questions to further explore this topic:

1. What is the quotient rule, and when is it used?
2. How would the derivative change if the function was \( \frac{\ln(x)}{\ln(x) - 3} \)?
3. What happens to the derivative if we add a constant outside the fraction, say \( f(x) = \frac{\ln(x)}{\ln(x) + 3} + 5 \)?
4. How can we find the derivative of the inverse function of \( f(x) = \frac{\ln(x)}{\ln(x) + 3} \)?
5. How does logarithmic differentiation work, and could it apply here?

**Tip:** When dealing with logarithmic functions, always remember to use the chain rule and quotient rule carefully as they often simplify complex expressions effectively.

Find the following derivative: d/dx (ln(x) / (ln(x) + 3))

Learn how to find the derivative of ln(x) / (ln(x) + 3) using the quotient rule in calculus. This problem demonstrates key concepts in differentiation and logarithmic functions, suitable for high school and college-level calculus students.

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