Function Derivative 37E893

1. **State the problem:** We are given the function $$f(x) = \frac{x^3 + 7x^2 + x - 5}{x}$$ and asked to rewrite it by dividing each term in the numerator by the denominator, then simplify. After that, find the derivative $$f'(x)$$ of the simplified function. 2. **Rewrite the function:** Divide each term in the numerator by $$x$$: $$f(x) = \frac{x^3}{x} + \frac{7x^2}{x} + \frac{x}{x} - \frac{5}{x}$$ 3. **Simplify each term:** $$f(x) = x^{3-1} + 7x^{2-1} + x^{1-1} - 5x^{-1} = x^2 + 7x + 1 - 5x^{-1}$$ 4. **Final simplified function:** $$f(x) = x^2 + 7x + 1 - 5x^{-1}$$ 5. **Find the derivative:** Use the power rule $$\frac{d}{dx} x^n = nx^{n-1}$$ and constant rule. $$f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(7x) + \frac{d}{dx}(1) - \frac{d}{dx}(5x^{-1})$$ 6. **Calculate each derivative:** $$\frac{d}{dx}(x^2) = 2x$$ $$\frac{d}{dx}(7x) = 7$$ $$\frac{d}{dx}(1) = 0$$ $$\frac{d}{dx}(5x^{-1}) = 5 \cdot (-1) x^{-2} = -5x^{-2}$$ 7. **Combine results:** $$f'(x) = 2x + 7 + 0 - (-5x^{-2}) = 2x + 7 + 5x^{-2}$$ 8. **Final derivative:** $$f'(x) = 2x + 7 + \frac{5}{x^2}$$