To find the derivative of the expression \( f(x) = \frac{3 - x^2}{2 - x} \), we will use the **quotient rule**. The quotient rule is:

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

Here, let:
- \( u(x) = 3 - x^2 \)
- \( v(x) = 2 - x \)

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

- \( u'(x) = \frac{d}{dx} (3 - x^2) = -2x \)
- \( v'(x) = \frac{d}{dx} (2 - x) = -1 \)

### Step 2: Apply the quotient rule

Using the quotient rule formula:

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

### Step 3: Simplify the expression

1. Expand the numerator:

\[
(-2x)(2 - x) = -4x + 2x^2
\]
\[
(3 - x^2)(-1) = -3 + x^2
\]

So, the numerator becomes:

\[
(-4x + 2x^2) - (-3 + x^2) = -4x + 2x^2 + 3 - x^2 = x^2 - 4x + 3
\]

2. The denominator remains:

\[
(2 - x)^2 = (2 - x)(2 - x) = 4 - 4x + x^2
\]

### Final answer:

The derivative is:

\[
f'(x) = \frac{x^2 - 4x + 3}{x^2 - 4x + 4}
\]

Let me know if you'd like more details or have any questions!

Here are 5 related questions:
1. How would you find the derivative of a product of two functions?
2. Can you simplify the final derivative further for specific values of \(x\)?
3. What happens to the derivative if \(x = 2\) in this function?
4. How would the quotient rule change if both numerator and denominator had higher powers of \(x\)?
5. What is the second derivative of the given function?

**Tip:** Always simplify expressions before applying differentiation rules to avoid unnecessary complexity.

Learn how to find the derivative of the function (3 - x^2) / (2 - x) using the quotient rule. This calculus problem explores key concepts in differentiation, including the application of the quotient rule and simplification techniques. Suitable for students in Grades 11-12.

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