Browse Questions

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

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