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To differentiate the function \( g(x) = \ln\left( \frac{7 - x}{7 + x} \right) \), we can use the chain rule and the derivative properties of the natural logarithm.

### Steps:
1. **Let** \( u = \frac{7 - x}{7 + x} \).
2. Then, \( g(x) = \ln(u) \).
3. The derivative of \( g(x) \) with respect to \( x \) is given by \( g'(x) = \frac{1}{u} \cdot \frac{du}{dx} \).

Now we need to find \( \frac{du}{dx} \).

### Finding \( \frac{du}{dx} \):
\[ u = \frac{7 - x}{7 + x} \]
We can use the quotient rule, where if \( u = \frac{f(x)}{g(x)} \), then:
\[ \frac{du}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

Here, \( f(x) = 7 - x \) and \( g(x) = 7 + x \).

- \( f'(x) = -1 \)
- \( g'(x) = 1 \)

Applying the quotient rule:
\[
\frac{du}{dx} = \frac{(-1)(7 + x) - (7 - x)(1)}{(7 + x)^2} = \frac{-7 - x - 7 + x}{(7 + x)^2} = \frac{-14}{(7 + x)^2}
\]

### Now, substituting back:
\[
g'(x) = \frac{1}{\frac{7 - x}{7 + x}} \cdot \frac{-14}{(7 + x)^2}
\]

Simplify further:
\[
g'(x) = \frac{7 + x}{7 - x} \cdot \frac{-14}{(7 + x)^2}
\]

\[
g'(x) = \frac{-14}{(7 - x)(7 + x)}
\]

Finally, we get:
\[
g'(x) = \frac{-14}{49 - x^2}
\]

So, the derivative \( g'(x) = \frac{-14}{49 - x^2} \).

Do you want more details or have any questions? 

**Related Questions:**
1. How do you find the derivative of a logarithmic function involving a different base?
2. What is the chain rule in calculus, and when is it used?
3. How can we differentiate a function involving trigonometric expressions?
4. What are the common properties of logarithms useful in differentiation?
5. How would you integrate the function \( g'(x) \) to find the original \( g(x) \)?

**Tip:** Always simplify the argument inside the logarithm before differentiating, if possible, to make the differentiation process easier.

Learn how to differentiate the function g(x) = ln((7-x)/(7+x)) using the chain rule and quotient rule. This solution covers key calculus concepts including natural logarithms, differentiation techniques, and step-by-step application of the quotient rule. Suitable for undergraduate calculus students.

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