1. **State the problem:** We are given the function $$f(x) = (4x^3 - 2x^2 + 3x + 1)(x^{-2} - \sqrt{x})$$ and asked to analyze it using basic calculus. 2. **Rewrite the function:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the function becomes: $$f(x) = (4x^3 - 2x^2 + 3x + 1)(x^{-2} - x^{\frac{1}{2}})$$ 3. **Find the derivative using the product rule:** The product rule states: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $$u(x) = 4x^3 - 2x^2 + 3x + 1$$ and $$v(x) = x^{-2} - x^{\frac{1}{2}}$$ 4. **Calculate derivatives of each part:** $$u'(x) = \frac{d}{dx}(4x^3) - \frac{d}{dx}(2x^2) + \frac{d}{dx}(3x) + \frac{d}{dx}(1) = 12x^2 - 4x + 3 + 0 = 12x^2 - 4x + 3$$ $$v'(x) = \frac{d}{dx}(x^{-2}) - \frac{d}{dx}(x^{\frac{1}{2}}) = -2x^{-3} - \frac{1}{2}x^{-\frac{1}{2}}$$ 5. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = (12x^2 - 4x + 3)(x^{-2} - x^{\frac{1}{2}}) + (4x^3 - 2x^2 + 3x + 1)(-2x^{-3} - \frac{1}{2}x^{-\frac{1}{2}})$$ 6. **Simplify each term:** First term: $$ (12x^2 - 4x + 3)(x^{-2} - x^{\frac{1}{2}}) = (12x^2)(x^{-2}) - (12x^2)(x^{\frac{1}{2}}) - (4x)(x^{-2}) + (4x)(x^{\frac{1}{2}}) + 3(x^{-2}) - 3(x^{\frac{1}{2}}) $$ Simplify powers: $$ 12x^{2-2} - 12x^{2+\frac{1}{2}} - 4x^{1-2} + 4x^{1+\frac{1}{2}} + 3x^{-2} - 3x^{\frac{1}{2}} = 12x^0 - 12x^{\frac{5}{2}} - 4x^{-1} + 4x^{\frac{3}{2}} + 3x^{-2} - 3x^{\frac{1}{2}} $$ Second term: $$ (4x^3 - 2x^2 + 3x + 1)(-2x^{-3} - \frac{1}{2}x^{-\frac{1}{2}}) = -2(4x^3)x^{-3} + 2(2x^2)x^{-3} - 2(3x)x^{-3} - 2(1)x^{-3} - \frac{1}{2}(4x^3)x^{-\frac{1}{2}} + \frac{1}{2}(2x^2)x^{-\frac{1}{2}} - \frac{1}{2}(3x)x^{-\frac{1}{2}} - \frac{1}{2}(1)x^{-\frac{1}{2}} $$ Simplify powers: $$ -8x^{3-3} + 4x^{2-3} - 6x^{1-3} - 2x^{-3} - 2x^{3-\frac{1}{2}} + x^{2-\frac{1}{2}} - \frac{3}{2}x^{1-\frac{1}{2}} - \frac{1}{2}x^{-\frac{1}{2}} $$ Which is: $$ -8x^0 + 4x^{-1} - 6x^{-2} - 2x^{-3} - 2x^{\frac{5}{2}} + x^{\frac{3}{2}} - \frac{3}{2}x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{1}{2}} $$ 7. **Combine like terms:** $$f'(x) = (12 - 8) + (-12x^{\frac{5}{2}} - 2x^{\frac{5}{2}}) + (-4x^{-1} + 4x^{-1}) + (4x^{\frac{3}{2}} + x^{\frac{3}{2}}) + (3x^{-2} - 6x^{-2}) + (-3x^{\frac{1}{2}} - \frac{3}{2}x^{\frac{1}{2}}) - \frac{1}{2}x^{-\frac{1}{2}} - 2x^{-3} $$ Simplify coefficients: $$4 - 14x^{\frac{5}{2}} + 0 + 5x^{\frac{3}{2}} - 3x^{-2} - \frac{9}{2}x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{1}{2}} - 2x^{-3}$$ 8. **Final derivative:** $$\boxed{f'(x) = 4 - 14x^{\frac{5}{2}} + 5x^{\frac{3}{2}} - 3x^{-2} - \frac{9}{2}x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{1}{2}} - 2x^{-3}}$$ This derivative tells us the rate of change of the function at any point $x$ where it is defined. **Note:** The function and derivative are defined only for $x > 0$ because of the square root and negative powers.