The problem asks us to differentiate the function $f(x) = \frac{x^3 - 2x^2 + x - 3}{x}$ with respect to $x$.

1. Problem Description

The problem asks us to differentiate the function $f(x) = \frac{x^3 - 2x^2 + x - 3}{x}$ with respect to $x$.

2. Solution Steps

First, we simplify the expression by dividing each term in the numerator by $x$:

$f(x) = \frac{x^3}{x} - \frac{2x^2}{x} + \frac{x}{x} - \frac{3}{x}$

$f(x) = x^2 - 2x + 1 - 3x^{-1}$

Next, we differentiate each term with respect to $x$.

Recall the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$.

$\frac{d}{dx}(x^2) = 2x^{2-1} = 2x$

$\frac{d}{dx}(-2x) = -2\frac{d}{dx}(x) = -2(1)x^{1-1} = -2(1) = -2$

$\frac{d}{dx}(1) = 0$

$\frac{d}{dx}(-3x^{-1}) = -3\frac{d}{dx}(x^{-1}) = -3(-1)x^{-1-1} = 3x^{-2} = \frac{3}{x^2}$

Combining these results, we get:

$f'(x) = 2x - 2 + 0 + \frac{3}{x^2}$

$f'(x) = 2x - 2 + \frac{3}{x^2}$

3. Final Answer

$2x - 2 + \frac{3}{x^2}$