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To find the derivative of the given function:

\[
f(x) = \ln\left( \frac{6x}{x + 9} \right)
\]

We will apply the chain rule and quotient rule step by step.

### Step 1: Use the logarithmic derivative rule
The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u \) is the inside function. In this case:

\[
u(x) = \frac{6x}{x + 9}
\]

So we will first need the derivative of \( u(x) \).

### Step 2: Differentiate \( u(x) = \frac{6x}{x + 9} \) using the quotient rule
The quotient rule states:

\[
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
\]

Here, \( f(x) = 6x \) and \( g(x) = x + 9 \).

- The derivative of \( f(x) = 6x \) is \( f'(x) = 6 \).
- The derivative of \( g(x) = x + 9 \) is \( g'(x) = 1 \).

Now apply the quotient rule:

\[
u'(x) = \frac{6(x + 9) - 6x(1)}{(x + 9)^2} = \frac{6x + 54 - 6x}{(x + 9)^2} = \frac{54}{(x + 9)^2}
\]

### Step 3: Apply the chain rule
Now, use the derivative of the natural logarithm function:

\[
f'(x) = \frac{1}{u(x)} \cdot u'(x)
\]

Substitute \( u(x) = \frac{6x}{x + 9} \) and \( u'(x) = \frac{54}{(x + 9)^2} \):

\[
f'(x) = \frac{1}{\frac{6x}{x + 9}} \cdot \frac{54}{(x + 9)^2} = \frac{x + 9}{6x} \cdot \frac{54}{(x + 9)^2}
\]

Simplify the expression:

\[
f'(x) = \frac{54}{6x(x + 9)} = \frac{9}{x(x + 9)}
\]

Thus, the derivative of the function is:

\[
f'(x) = \frac{9}{x(x + 9)}
\]

Would you like further details or have any questions?

Here are some related questions for practice:
1. How would you differentiate a more complex logarithmic function?
2. Can you find the second derivative of \( f(x) = \ln\left( \frac{6x}{x + 9} \right) \)?
3. How would you apply logarithmic differentiation to a function with powers of \( x \)?
4. What is the significance of the chain rule when differentiating composite functions?
5. How does the quotient rule differ from the product rule?

**Tip:** The chain rule is essential when differentiating functions that involve composition, so always identify the "inner" and "outer" functions first.

Find the derivative of the function: f(x) = ln(6x / (x + 9))

Learn how to find the derivative of f(x) = ln(6x / (x + 9)) using logarithmic differentiation, the quotient rule, and chain rule. Detailed step-by-step solution provided for high school and college-level calculus students.

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