To find the first derivative of the expression:

\[
f(x) = \sqrt{\frac{1 + x}{1 - x}},
\]

we will apply the chain rule and the quotient rule.

### Step-by-Step Solution

1. **Rewrite the expression**:

   Let:

   \[
   u(x) = \frac{1 + x}{1 - x}.
   \]

   Then:

   \[
   f(x) = \sqrt{u(x)} = (u(x))^{1/2}.
   \]

2. **Find the derivative using the chain rule**:

   The derivative of \(f(x)\) with respect to \(x\) is:

   \[
   f'(x) = \frac{d}{dx} \left[ (u(x))^{1/2} \right] = \frac{1}{2} (u(x))^{-1/2} \cdot u'(x).
   \]

   Now, substitute \(u(x) = \frac{1 + x}{1 - x}\):

   \[
   f'(x) = \frac{1}{2 \sqrt{\frac{1 + x}{1 - x}}} \cdot u'(x).
   \]

3. **Find the derivative \(u'(x)\) using the quotient rule**:

   The quotient rule states:

   \[
   \frac{d}{dx} \left( \frac{A}{B} \right) = \frac{A'B - AB'}{B^2}.
   \]

   Here, \(A = 1 + x\) and \(B = 1 - x\). So:

   \[
   A' = 1, \quad B' = -1.
   \]

   Applying the quotient rule:

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

4. **Substitute \(u'(x)\) back into the derivative \(f'(x)\)**:

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

5. **Simplify the derivative**:

   The expression simplifies to:

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

   Simplifying further:

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

   Combining like terms:

   \[
   f'(x) = \frac{\sqrt{1 - x}}{(1 - x)^{3/2} (1 + x)^{1/2}} = \frac{1}{\sqrt{(1 - x)^3 (1 + x)}}.
   \]

So, the first derivative of the function is:

\[
\boxed{f'(x) = \frac{1}{\sqrt{(1 - x)^3 (1 + x)}}.}
\]

Would you like a deeper explanation, or do you have any other questions about this? 

Here are some related questions:
1. How do you find the second derivative of this function?
2. What is the domain of \(f'(x)\)?
3. How can you use the derivative to find the critical points of the function?
4. How do you check if the function has any points of inflection?
5. What is the behavior of \(f(x)\) as \(x \to 1\) or \(x \to -1\)?

**Tip:** When taking derivatives, always remember to consider the chain rule if the function is composed of multiple layers!

Learn how to find the first derivative of the function f(x) = sqrt((1 + x) / (1 - x)) step-by-step using the chain rule and quotient rule. This problem involves differentiation techniques and is suitable for advanced high school students and college calculus students.

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