Find the derivative of the following function with respect to $x$ (where $p, q, r, s$ are fixed non-zero constants): $(p x+q)\left(\frac{r}{x}+s\right)$.

Let $f(x) = (p x+q)\left(\frac{r}{x}+s\right)$.
Using the product rule,$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$:
$f'(x) = (p x+q) \frac{d}{dx}\left(\frac{r}{x}+s\right) + \left(\frac{r}{x}+s\right) \frac{d}{dx}(p x+q)$
$f'(x) = (p x+q) \left(-\frac{r}{x^2}\right) + \left(\frac{r}{x}+s\right) (p)$
$f'(x) = -\frac{p r x}{x^2} - \frac{q r}{x^2} + \frac{p r}{x} + p s$
$f'(x) = -\frac{p r}{x} - \frac{q r}{x^2} + \frac{p r}{x} + p s$
$f'(x) = p s - \frac{q r}{x^2}$