1. **Problem:** Find the derivative of $$y = \frac{x^5 + 57x^2}{5x^3 - 6x}$$ using the quotient rule. 2. **Formula:** The quotient rule states: $$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$ where $$u = x^5 + 57x^2$$ and $$v = 5x^3 - 6x$$. 3. **Find derivatives of $$u$$ and $$v$$:** $$\frac{du}{dx} = 5x^4 + 114x$$ $$\frac{dv}{dx} = 15x^2 - 6$$ 4. **Apply quotient rule:** $$\frac{dy}{dx} = \frac{(5x^3 - 6x)(5x^4 + 114x) - (x^5 + 57x^2)(15x^2 - 6)}{(5x^3 - 6x)^2}$$ 5. **Simplify numerator:** Expand terms: $$(5x^3)(5x^4) = 25x^7$$ $$(5x^3)(114x) = 570x^4$$ $$(-6x)(5x^4) = -30x^5$$ $$(-6x)(114x) = -684x^2$$ Sum first part: $$25x^7 + 570x^4 - 30x^5 - 684x^2$$ Expand second part: $$(x^5)(15x^2) = 15x^7$$ $$(x^5)(-6) = -6x^5$$ $$(57x^2)(15x^2) = 855x^4$$ $$(57x^2)(-6) = -342x^2$$ Sum second part: $$15x^7 - 6x^5 + 855x^4 - 342x^2$$ 6. **Subtract second part from first:** $$[25x^7 + 570x^4 - 30x^5 - 684x^2] - [15x^7 - 6x^5 + 855x^4 - 342x^2]$$ $$= (25x^7 - 15x^7) + (570x^4 - 855x^4) + (-30x^5 + 6x^5) + (-684x^2 + 342x^2)$$ $$= 10x^7 - 285x^4 - 24x^5 - 342x^2$$ 7. **Rearrange terms:** $$10x^7 - 24x^5 - 285x^4 - 342x^2$$ 8. **Final derivative:** $$\frac{dy}{dx} = \frac{10x^7 - 24x^5 - 285x^4 - 342x^2}{(5x^3 - 6x)^2}$$ --- **Slug:** derivative_quotient **Subject:** calculus **Desmos:** {"latex": "y=\frac{x^5 + 57x^2}{5x^3 - 6x}", "features": {"intercepts": true, "extrema": true}} **q_count:** 1