Find the derivative of the following functions (it is to be understood that $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $(p x + q) (\frac{r}{x} + s)$

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Solution

Let $f (x) = (p x + q) (\frac{r}{x} + s)$
Thus using product rule
$f^{'} (x) = (p x + q) {(\frac{r}{x} + s)}^{'} + (\frac{r}{x} + s) (p x + q)^{'}$
$= (p x + q) {(r x^{- 1} + s)}^{'} + (\frac{r}{x} + s) (P)$
$= (p x + q) (- r x^{- 2}) + (\frac{r}{x} + s) p$
$= (p x + q) (\frac{- r}{x^{2}}) + (\frac{r}{x} + s) p$
$= \frac{- p r}{x} - \frac{q r}{x^{2}} + \frac{p r}{x} + p s = p s - \frac{q r}{x^{2}}$

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