Question

Find the derivative of $(px+q)(\frac{r}{x}+s)$ where $p,q,r$ and $s$ are fixed non-zero constants.

Answer 1

Let $f\left(x\right)=(px+q)(\frac{r}{x}+s)$
$\Rightarrow f\left(x\right)=pr+psx+\frac{qr}{x}+qs$
$\Rightarrow f\left(x\right)=psx+\frac{qr}{x}+qs+pr$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=ps-\frac{qr}{x^2}+0$
$\therefore f^{\prime }\left(x\right)=ps-\frac{rq}{x^2}$