Question

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) : $(px+q)(\frac{r}{x}+s)$

Answer 1

Let $f\left(x\right)=(px+q)(\frac{r}{x}+s)$
Thus using product rule
$f^{\prime }\left(x\right)=(px+q){(\frac{r}{x}+s)}^{\prime }+(\frac{r}{x}+s)(px+q{)}^{\prime }$
$=(px+q){(r{x}^{-1}+s)}^{\prime }+(\frac{r}{x}+s)\left(P\right)$
$=(px+q)(-r{x}^{-2})+(\frac{r}{x}+s)p$
$=(px+q)\left(\frac{-r}{x^2}\right)+(\frac{r}{x}+s)p$
$=\frac{-pr}{x}-\frac{qr}{x^2}+\frac{pr}{x}+ps=ps-\frac{qr}{x^2}$