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We are tasked with differentiating the function \( g(x) = \ln \left( \frac{8 - x}{8 + x} \right) \).

Let's go through the steps:

### Step 1: Recognize the structure
We have a natural logarithm, so we can apply the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u'(x) \).

Here, \( u(x) = \frac{8 - x}{8 + x} \).

### Step 2: Differentiate the inner function \( u(x) \)
Now, let's find the derivative of \( u(x) = \frac{8 - x}{8 + x} \) using the quotient rule. The quotient rule states:

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

Here:
- \( f(x) = 8 - x \), so \( f'(x) = -1 \),
- \( g(x) = 8 + x \), so \( g'(x) = 1 \).

Applying the quotient rule:

\[
u'(x) = \frac{(-1)(8 + x) - (8 - x)(1)}{(8 + x)^2}
\]
Simplify the numerator:

\[
u'(x) = \frac{-(8 + x) - (8 - x)}{(8 + x)^2} = \frac{-8 - x - 8 + x}{(8 + x)^2} = \frac{-16}{(8 + x)^2}
\]

### Step 3: Apply the chain rule
Now, use the chain rule to differentiate \( g(x) \):

\[
g'(x) = \frac{1}{\frac{8 - x}{8 + x}} \cdot \left( \frac{-16}{(8 + x)^2} \right)
\]

Simplify the expression:

\[
g'(x) = \frac{8 + x}{8 - x} \cdot \frac{-16}{(8 + x)^2}
\]

Cancel out one \( 8 + x \) term from the numerator and denominator:

\[
g'(x) = \frac{-16}{(8 - x)(8 + x)}
\]

Since \( (8 - x)(8 + x) = 64 - x^2 \), the final derivative is:

\[
g'(x) = \frac{-16}{64 - x^2}
\]

Thus, the derivative is:

\[
g'(x) = \frac{-16}{64 - x^2}
\]

Would you like further details on any step?

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Here are 5 questions you might find helpful to explore further:

1. How would you differentiate a more complex logarithmic function?
2. Can you apply the quotient rule to different types of rational functions?
3. How does the chain rule work when combining multiple functions?
4. What happens to the derivative when the denominator includes more complicated terms like trigonometric functions?
5. How can you simplify expressions after using the quotient rule efficiently?

**Tip:** When dealing with logarithmic derivatives, always simplify the inner function before differentiating to make the calculation easier.

Differentiate g(x) = ln((8 - x) / (8 + x)).

This guide explains the differentiation of g(x) = ln((8 - x) / (8 + x)) step by step. Learn how to apply the quotient rule and chain rule in logarithmic differentiation. Suitable for advanced high school or university students.

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