Question Number 203994 by Davidtim last updated on 03/Feb/24
$${d}\left(\frac{{x}!}{{dx}}\right)=? \\ $$
Answered by Frix last updated on 03/Feb/24
![∀x∈N: x!:=Π_(n=1) ^x n; 0!:=1 ⇒ ⇒ x! is not continuous ⇒ ((d[x!])/dx) does not exist.](https://www.tinkutara.com/question/Q204003.png)
$$\forall{x}\in\mathbb{N}:\:{x}!:=\underset{{n}=\mathrm{1}} {\overset{{x}} {\prod}}{n};\:\mathrm{0}!:=\mathrm{1}\:\Rightarrow \\ $$$$\Rightarrow\:{x}!\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous}\:\Rightarrow \\ $$$$\frac{{d}\left[{x}!\right]}{{dx}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist}. \\ $$
Answered by MathematicalUser2357 last updated on 06/Feb/24
$${x}!=\Gamma\left({x}+\mathrm{1}\right) \\ $$$$\frac{{d}\Gamma\left({x}+\mathrm{1}\right)}{{dx}}=\Gamma\left({x}+\mathrm{1}\right)\psi^{\left(\mathrm{0}\right)} \left({x}+\mathrm{1}\right) \\ $$