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Mathematics Question on Limits and derivations

\text{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

Let ,f(x)=(px+q)(rx+s)\text{Let }, f(x)=(px+q)(\frac{r}{x}+s)

By Leibnitz product rule,\text{By Leibnitz product rule,}

=f′(x)=(px+q)(rx+s)′+(rx+s)(px+q)′=f'(x)=(px+q)(\frac{r}{x}+s)'+(\frac{r}{x}+s)(px+q)'

=(px+q)(rx−1+s)′+(rs)(p)=(px+q)(rx^{-1}+s)'+(\frac{r}{s})(p)

=(px+q)(−rx−2)+(rs+s)p=(px+q)(-rx^{-2})+(\frac{r}{s}+s)p

=(px+q)(−rx2)+(rs)p=(px+q)(\frac{-r}{x^2})+(\frac{r}{s})p

=−prx−qrx2+prx+ps=-\frac{pr}{x}-\frac{qr}{x^2}+\frac{pr}{x}+ps

=ps−qrx2=ps-\frac{qr}{x^2}