To derive the function \( f(x) = \frac{3x + 1}{x - 2} \), we'll use the **quotient rule**, which states:

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

Here:
- \( u(x) = 3x + 1 \)
- \( v(x) = x - 2 \)

### Step 1: Find \( u'(x) \) and \( v'(x) \)
- \( u'(x) = 3 \)
- \( v'(x) = 1 \)

### Step 2: Apply the quotient rule
\[
f'(x) = \frac{(3)(x - 2) - (3x + 1)(1)}{(x - 2)^2}
\]

### Step 3: Simplify the expression
\[
f'(x) = \frac{3(x - 2) - (3x + 1)}{(x - 2)^2}
\]
\[
f'(x) = \frac{3x - 6 - 3x - 1}{(x - 2)^2}
\]
\[
f'(x) = \frac{-7}{(x - 2)^2}
\]

Thus, the derivative of \( f(x) = \frac{3x + 1}{x - 2} \) is:

\[
f'(x) = \frac{-7}{(x - 2)^2}
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. How do you find the derivative of more complex rational functions?
2. Can you derive a function using the product rule instead of the quotient rule?
3. What is the physical interpretation of a derivative in real-world contexts?
4. How do you handle higher-order derivatives of rational functions?
5. What are the conditions under which the quotient rule cannot be applied?

**Tip**: Always double-check for simplification opportunities when using derivative rules—this helps avoid calculation errors.

Explore how to derive the function (3x + 1) / (x - 2) step-by-step using the quotient rule. This solution covers essential calculus concepts like finding derivatives of rational functions and applying the quotient rule. Suitable for high school students in Grades 11-12.

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