Find the derivative of the following functions


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


Let $f(x)=(p x+q)\left(\frac{r}{x}+s\right)$

By Leibnitz product rule,

$f^{\prime}(x)=(p x+q)\left(\frac{r}{x}+s\right)^{\prime}+\left(\frac{r}{x}+s\right)(p x+q)^{\prime}$

$=(p x+q)\left(r x^{-1}+s\right)^{\prime}+\left(\frac{r}{x}+s\right)(p)$

$=(p x+q)\left(-r x^{-2}\right)+\left(\frac{r}{x}+s\right) p$

$=(p x+q)\left(\frac{-r}{x^{2}}\right)+\left(\frac{r}{x}+s\right) p$

$=(p x+q)\left(\frac{-r}{x^{2}}\right)+\left(\frac{r}{x}+s\right) p$

$=\frac{-p r}{x}-\frac{q r}{x^{2}}+\frac{p r}{x}+p s$

$=p s-\frac{q r}{x^{2}}$


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