Find the derivative of the following function (it is to be understood that $p, q, r$ are fixed non-zero constants): $\frac{p x^{2}+q x+r}{x}$

Let $f(x) = \frac{p x^{2}+q x+r}{x}$.
We can simplify the function by dividing each term in the numerator by $x$:
$f(x) = \frac{p x^{2}}{x} + \frac{q x}{x} + \frac{r}{x} = p x + q + r x^{-1}$.
Now,differentiate with respect to $x$:
$f'(x) = \frac{d}{dx}(p x) + \frac{d}{dx}(q) + \frac{d}{dx}(r x^{-1})$.
Using the power rule $\frac{d}{dx}(x^n) = n x^{n-1}$:
$f'(x) = p(1) + 0 + r(-1)x^{-2}$.
Therefore,$f'(x) = p - \frac{r}{x^{2}}$.