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Question-123124




Question Number 123124 by aupo14 last updated on 23/Nov/20
Commented by Dwaipayan Shikari last updated on 23/Nov/20
x!=Γ(x+1)  log(x!)=log(Γ(x+1))  (((dx!)/dx)/(x!))=((Γ′(x+1))/(Γ(x+1)))  ⇒(d/dx)(x!)=x!.ψ(x+1)
$${x}!=\Gamma\left({x}+\mathrm{1}\right) \\ $$$${log}\left({x}!\right)={log}\left(\Gamma\left({x}+\mathrm{1}\right)\right) \\ $$$$\frac{\frac{{dx}!}{{dx}}}{{x}!}=\frac{\Gamma'\left({x}+\mathrm{1}\right)}{\Gamma\left({x}+\mathrm{1}\right)} \\ $$$$\Rightarrow\frac{{d}}{{dx}}\left({x}!\right)={x}!.\psi\left({x}+\mathrm{1}\right) \\ $$$$ \\ $$

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