Question Number 42514 by jbm last updated on 27/Aug/18
$${calculate}\:\:{d}\left({x}!\right)/{dx}=\:? \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Aug/18
$${y}={x}! \\ $$$${y}={x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)...\mathrm{1} \\ $$$${lny}={lnx}+{ln}\left({x}−\mathrm{1}\right)+{ln}\left({x}−\mathrm{2}\right)+...+\mathrm{1} \\ $$$$\frac{\mathrm{1}}{{y}}\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}}+\frac{\mathrm{1}}{{x}−\mathrm{2}}+... \\ $$$$\frac{{dy}}{{dx}}={x}!\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}}+\frac{\mathrm{1}}{{x}−\mathrm{2}}+...\right) \\ $$