Question: Find $f'(x)$ for the function $f(x) = \frac{x + 8}{x^2 - 9x + 2}$.
1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = \frac{x + 8}{x^2 - 9x + 2}.$$
2. **Recall the formula:** For a function given by a quotient $$f(x) = \frac{u(x)}{v(x)},$$ the derivative is given by the quotient rule:
$$f'(x) = \frac{v(x) u'(x) - u(x) v'(x)}{[v(x)]^2}.$$
3. **Identify $u(x)$ and $v(x)$:**
$$u(x) = x + 8,$$
$$v(x) = x^2 - 9x + 2.$$
4. **Compute derivatives $u'(x)$ and $v'(x)$:**
$$u'(x) = 1,$$
$$v'(x) = 2x - 9.$$
5. **Apply the quotient rule:**
$$f'(x) = \frac{(x^2 - 9x + 2)(1) - (x + 8)(2x - 9)}{(x^2 - 9x + 2)^2}.$$
6. **Expand the numerator:**
$$= \frac{x^2 - 9x + 2 - [(x)(2x - 9) + 8(2x - 9)]}{(x^2 - 9x + 2)^2}.$$
7. **Distribute terms inside the brackets:**
$$= \frac{x^2 - 9x + 2 - [2x^2 - 9x + 16x - 72]}{(x^2 - 9x + 2)^2}.$$
8. **Simplify inside the brackets:**
$$= \frac{x^2 - 9x + 2 - (2x^2 + 7x - 72)}{(x^2 - 9x + 2)^2}.$$
9. **Subtract the terms:**
$$= \frac{x^2 - 9x + 2 - 2x^2 - 7x + 72}{(x^2 - 9x + 2)^2}.$$
10. **Combine like terms:**
$$= \frac{(x^2 - 2x^2) + (-9x - 7x) + (2 + 72)}{(x^2 - 9x + 2)^2} = \frac{-x^2 - 16x + 74}{(x^2 - 9x + 2)^2}.$$
**Final answer:**
$$\boxed{f'(x) = \frac{-x^2 - 16x + 74}{(x^2 - 9x + 2)^2}}.$$