To find the derivative of the given function using the quotient rule, we need to use the formula for the derivative of a quotient:

$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$

where $u = x^2 - 9x + 4$ and $v = x^2 + 9$.

Let's follow these steps:

1. Differentiate $u$ and $v$:

   • $u' = \frac{d}{dx}(x^2 - 9x + 4) = 2x - 9$
   • $v' = \frac{d}{dx}(x^2 + 9) = 2x$

2. Apply the quotient rule: $y' = \frac{(2x - 9)(x^2 + 9) - (x^2 - 9x + 4)(2x)}{(x^2 + 9)^2}$

3. Expand the numerator:

   • Expand $(2x - 9)(x^2 + 9)$: $(2x - 9)(x^2 + 9) = 2x(x^2) + 2x(9) - 9(x^2) - 9(9) = 2x^3 + 18x - 9x^2 - 81$
   • Expand $(x^2 - 9x + 4)(2x)$: $(x^2 - 9x + 4)(2x) = 2x(x^2) - 2x(9x) + 2x(4) = 2x^3 - 18x^2 + 8x$

4. Combine these expressions in the numerator: $y' = \frac{(2x^3 + 18x - 9x^2 - 81) - (2x^3 - 18x^2 + 8x)}{(x^2 + 9)^2}$

   Simplifying the numerator: $2x^3 + 18x - 9x^2 - 81 - 2x^3 + 18x^2 - 8x = 9x^2 + 10x - 81$

5. Final derivative: $y' = \frac{9x^2 + 10x - 81}{(x^2 + 9)^2}$

Let me know if you need more details or have any questions. Here are some related questions to further your understanding:

1. How do you derive the quotient rule from first principles?
2. What are some common mistakes to watch out for when using the quotient rule?
3. How does the quotient rule compare with the product rule in terms of usage?
4. Can you use the chain rule in conjunction with the quotient rule for more complex functions?
5. What are the practical applications of using derivatives in real-life situations?

Tip: When using the quotient rule, ensure you carefully differentiate both the numerator and denominator before substituting them into the formula to avoid errors.