1. **Problem:** Find the derivative of $$\frac{x^5 + 57x^2}{5x^3 - 6x}$$. 2. **Formula:** Use the quotient rule: $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$ where $$u = x^5 + 57x^2$$ and $$v = 5x^3 - 6x$$. 3. **Calculate derivatives:** $$\frac{du}{dx} = 5x^4 + 114x$$ $$\frac{dv}{dx} = 15x^2 - 6$$ 4. **Apply quotient rule:** $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{(5x^3 - 6x)(5x^4 + 114x) - (x^5 + 57x^2)(15x^2 - 6)}{(5x^3 - 6x)^2}$$ 5. **Simplify numerator:** Expand terms: $$(5x^3)(5x^4) = 25x^7$$ $$(5x^3)(114x) = 570x^4$$ $$( -6x)(5x^4) = -30x^5$$ $$( -6x)(114x) = -684x^2$$ $$(x^5)(15x^2) = 15x^7$$ $$(x^5)(-6) = -6x^5$$ $$(57x^2)(15x^2) = 855x^4$$ $$(57x^2)(-6) = -342x^2$$ Combine: $$25x^7 + 570x^4 - 30x^5 - 684x^2 - (15x^7 - 6x^5 + 855x^4 - 342x^2)$$ $$= 25x^7 + 570x^4 - 30x^5 - 684x^2 - 15x^7 + 6x^5 - 855x^4 + 342x^2$$ $$= (25x^7 - 15x^7) + (570x^4 - 855x^4) + (-30x^5 + 6x^5) + (-684x^2 + 342x^2)$$ $$= 10x^7 - 285x^4 - 24x^5 - 342x^2$$ 6. **Final derivative:** $$\frac{d}{dx} = \frac{10x^7 - 24x^5 - 285x^4 - 342x^2}{(5x^3 - 6x)^2}$$ --- 1. **Problem:** Find the derivative of $$\frac{4x - 4 + 7x^2}{-5x + x^2}$$. 2. **Formula:** Use quotient rule with $$u = 4x - 4 + 7x^2$$ and $$v = -5x + x^2$$. 3. **Derivatives:** $$\frac{du}{dx} = 4 + 14x$$ $$\frac{dv}{dx} = -5 + 2x$$ 4. **Apply quotient rule:** $$\frac{d}{dx} = \frac{(-5x + x^2)(4 + 14x) - (4x - 4 + 7x^2)(-5 + 2x)}{(-5x + x^2)^2}$$ 5. **Simplify numerator:** Expand: $$(-5x)(4) = -20x$$ $$(-5x)(14x) = -70x^2$$ $$(x^2)(4) = 4x^2$$ $$(x^2)(14x) = 14x^3$$ $$(4x)(-5) = -20x$$ $$(4x)(2x) = 8x^2$$ $$(-4)(-5) = 20$$ $$(-4)(2x) = -8x$$ $$(7x^2)(-5) = -35x^2$$ $$(7x^2)(2x) = 14x^3$$ Combine: $$( -20x - 70x^2 + 4x^2 + 14x^3 ) - ( -20x + 8x^2 + 20 - 8x - 35x^2 + 14x^3 )$$ $$= (-20x - 66x^2 + 14x^3) - (-28x - 27x^2 + 14x^3 + 20)$$ $$= -20x - 66x^2 + 14x^3 + 28x + 27x^2 - 14x^3 - 20$$ $$= ( -20x + 28x ) + ( -66x^2 + 27x^2 ) + (14x^3 - 14x^3) - 20$$ $$= 8x - 39x^2 - 20$$ 6. **Final derivative:** $$\frac{d}{dx} = \frac{8x - 39x^2 - 20}{(-5x + x^2)^2}$$ --- 1. **Problem:** Find the derivative of $$\frac{\frac{2}{3}x (x^4 + \frac{1}{2}x^2)}{5x^5 - 6\sqrt{x}}$$. 2. **Rewrite numerator:** $$u = \frac{2}{3}x (x^4 + \frac{1}{2}x^2) = \frac{2}{3}x^5 + \frac{1}{3}x^3$$ 3. **Denominator:** $$v = 5x^5 - 6x^{1/2}$$ 4. **Derivatives:** $$\frac{du}{dx} = \frac{2}{3} \cdot 5x^4 + \frac{1}{3} \cdot 3x^2 = \frac{10}{3}x^4 + x^2$$ $$\frac{dv}{dx} = 25x^4 - 3x^{-1/2}$$ 5. **Apply quotient rule:** $$\frac{d}{dx} = \frac{(5x^5 - 6x^{1/2})(\frac{10}{3}x^4 + x^2) - (\frac{2}{3}x^5 + \frac{1}{3}x^3)(25x^4 - 3x^{-1/2})}{(5x^5 - 6x^{1/2})^2}$$ 6. **Final derivative:** Expression above is the derivative; further simplification is possible but lengthy. --- 1. **Problem:** Find the derivative of $$\frac{5x^5 - 6\sqrt{x}}{(5x^5 - 6\sqrt{x}) x^{1/2}}$$. 2. **Rewrite denominator:** $$v = (5x^5 - 6x^{1/2}) x^{1/2} = 5x^5 x^{1/2} - 6x^{1/2} x^{1/2} = 5x^{11/2} - 6x$$ 3. **Numerator:** $$u = 5x^5 - 6x^{1/2}$$ 4. **Derivatives:** $$\frac{du}{dx} = 25x^4 - 3x^{-1/2}$$ $$\frac{dv}{dx} = \frac{d}{dx}(5x^{11/2} - 6x) = \frac{55}{2}x^{9/2} - 6$$ 5. **Apply quotient rule:** $$\frac{d}{dx} = \frac{(5x^{11/2} - 6x)(25x^4 - 3x^{-1/2}) - (5x^5 - 6x^{1/2})(\frac{55}{2}x^{9/2} - 6)}{(5x^{11/2} - 6x)^2}$$ --- 1. **Problem:** Find the derivative of $$\frac{x - 6 + 57x^2}{5x^{1/3} - 6\sqrt[3]{3x}}$$. 2. **Rewrite denominator:** $$v = 5x^{1/3} - 6(3x)^{1/3} = 5x^{1/3} - 6 \cdot 3^{1/3} x^{1/3} = (5 - 6 \cdot 3^{1/3}) x^{1/3}$$ 3. **Numerator:** $$u = x - 6 + 57x^2$$ 4. **Derivatives:** $$\frac{du}{dx} = 1 + 114x$$ $$\frac{dv}{dx} = (5 - 6 \cdot 3^{1/3}) \cdot \frac{1}{3} x^{-2/3}$$ 5. **Apply quotient rule:** $$\frac{d}{dx} = \frac{v(1 + 114x) - u \cdot (5 - 6 \cdot 3^{1/3}) \cdot \frac{1}{3} x^{-2/3}}{v^2}$$ --- **Summary:** Each derivative uses the quotient rule with careful differentiation of numerator and denominator functions.