1. **Derivatives** **Problem:** Find the derivatives of the given functions. **Formulas:** - Power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ - Sum rule: $$\frac{d}{dx} (f + g) = f' + g'$$ - Product rule: $$\frac{d}{dx} (fg) = f'g + fg'$$ **Step 1:** Derivative of $$y = -8x^{-4} + 8x^{-3} - x$$ Apply the power rule to each term: $$\frac{dy}{dx} = -8 \cdot (-4) x^{-4-1} + 8 \cdot (-3) x^{-3-1} - 1 \cdot x^{1-1}$$ $$= 32 x^{-5} - 24 x^{-4} - 1$$ **Step 2:** Derivative of $$y = (x^7 + 1)(x^2 - 3)$$ Use the product rule: Let $$f = x^7 + 1$$ and $$g = x^2 - 3$$ $$f' = 7x^6$$ $$g' = 2x$$ Then, $$\frac{dy}{dx} = f'g + fg' = 7x^6 (x^2 - 3) + (x^7 + 1)(2x)$$ Expand: $$= 7x^8 - 21x^6 + 2x^8 + 2x$$ Combine like terms: $$= (7x^8 + 2x^8) - 21x^6 + 2x = 9x^8 - 21x^6 + 2x$$ 2. **Integrals** **Problem:** Find the indefinite integrals. **Formula:** - Power rule for integrals: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, n \neq -1$$ - Constant multiple rule: $$\int a f(x) dx = a \int f(x) dx$$ - Sum rule: $$\int (f + g) dx = \int f dx + \int g dx$$ **Step 1:** $$\int x dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C$$ **Step 2:** $$\int -ax dx = -a \int x dx = -a \cdot \frac{x^2}{2} + C = -\frac{a x^2}{2} + C$$ **Step 3:** $$\int (2x - 5) dx = 2 \int x dx - 5 \int dx = 2 \cdot \frac{x^2}{2} - 5x + C = x^2 - 5x + C$$ **Step 4:** $$\int (x^4 + x^3 + x^2 - x - 6) dx = \frac{x^{5}}{5} + \frac{x^{4}}{4} + \frac{x^{3}}{3} - \frac{x^{2}}{2} - 6x + C$$ **Step 5:** $$\int 7x^6 dx = 7 \cdot \frac{x^{7}}{7} + C = x^{7} + C$$ **Step 6:** $$\int 12 dy = 12y + C$$ **Step 7:** $$\int (5x^4 + 4x^3 - 3x^2 + 2x - 6) dx = \frac{5x^{5}}{5} + \frac{4x^{4}}{4} - \frac{3x^{3}}{3} + \frac{2x^{2}}{2} - 6x + C = x^{5} + x^{4} - x^{3} + x^{2} - 6x + C$$ **Step 8:** $$\int (-5x^{-6} + 4x^{-5} - 3x^{-4} - 2x^{-3} - x^{-2} + 20) dx$$ Integrate term by term: $$= -5 \cdot \frac{x^{-5}}{-5} + 4 \cdot \frac{x^{-4}}{-4} - 3 \cdot \frac{x^{-3}}{-3} - 2 \cdot \frac{x^{-2}}{-2} - 1 \cdot \frac{x^{-1}}{-1} + 20x + C$$ Simplify: $$= x^{-5} - x^{-4} + x^{-3} + x^{-2} + x^{-1} + 20x + C$$ **Final answers:** 1) $$\frac{dy}{dx} = 32 x^{-5} - 24 x^{-4} - 1$$ 2) $$\frac{dy}{dx} = 9x^{8} - 21x^{6} + 2x$$ Integrals as above.