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

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