Let $f(x) = (px+q) \bigg( \large\frac{r}{x}+s \bigg)$

By Leibnitz product rule

$f'(x) = (px+q) \bigg( \large\frac{r}{x}+s \bigg)'$$\bigg( \large\frac{r}{x}+s \bigg)$$(px+q)'$

$ = (px+q)(rx^{-1}+s)'+ \bigg( \large\frac{r}{x}+s \bigg)(p)$

$ = (px+q)(-rx^{-2})+ \bigg( \large\frac{r}{x}+s \bigg)p$

$ = (px+q)\bigg( \large\frac{-r}{x^2}\bigg)$$+\bigg( \large\frac{r}{x}+s \bigg)p$

$ = \large\frac{-pr}{x}$$ - \large\frac{qr}{x^2}$$+\large\frac{pr}{x}$$+ps$

$ = ps - \large\frac{qr}{x^2}$