Question: Differentiate the function $y = -3(x^2 - 2) / (5x + 2)$ using the quotient rule.
1. **State the problem:** Differentiate the function $$y = -3\frac{x^2 - 2}{5x + 2}$$ using the quotient rule.
2. **Recall the quotient rule formula:** For a function $$y = \frac{u}{v}$$, the derivative is $$y' = \frac{v u' - u v'}{v^2}$$.
3. **Identify components:** Let $$u = -3(x^2 - 2) = -3x^2 + 6$$ and $$v = 5x + 2$$.
4. **Find derivatives:**
- $$u' = \frac{d}{dx}(-3x^2 + 6) = -6x$$
- $$v' = \frac{d}{dx}(5x + 2) = 5$$
5. **Apply quotient rule:**
$$y' = \frac{(5x + 2)(-6x) - (-3x^2 + 6)(5)}{(5x + 2)^2}$$
6. **Simplify numerator:**
$$= \frac{-30x^2 - 12x + 15x^2 - 30}{(5x + 2)^2}$$
7. **Combine like terms:**
$$= \frac{(-30x^2 + 15x^2) - 12x - 30}{(5x + 2)^2} = \frac{-15x^2 - 12x - 30}{(5x + 2)^2}$$
8. **Factor numerator if possible:**
$$-15x^2 - 12x - 30 = -3(5x^2 + 4x + 10)$$ (no simple factorization for quadratic inside).
9. **Final derivative:**
$$y' = \frac{-15x^2 - 12x - 30}{(5x + 2)^2}$$
This is the derivative of the given function using the quotient rule.