Question Number 17634 by tawa tawa last updated on 08/Jul/17
$$\mathrm{y}\:=\:\mathrm{x}!\:\:,\:\:\:\:\:\mathrm{Find}\:\:\:\mathrm{y}' \\ $$
Answered by alex041103 last updated on 09/Jul/17
$${Because}\:{x}!\:{is}\:{defined}\:{only}\:{for}\:{non}−{negative} \\ $$$${integers},\:{i}.{e}.\:{the}\:{function}\:{isn}'{t}\:{continious} \\ $$$${we}\:{cannot}\:{intagrate}\:{or}\:{diferentiate}\:{it}. \\ $$$${There}\:{is}\:{ananalitic}\:{continuation} \\ $$$${of}\:\:{the}\:{factorial}\:{function}\:{called}\:{the} \\ $$$${Gamma}\:{function} \\ $$$${x}!=\Gamma\left({x}+\mathrm{1}\right)=\underset{\:\:\mathrm{0}} {\overset{\infty} {\int}}{t}^{{x}} {e}^{−{t}} {dt} \\ $$$${You}\:{can}\:{search}\:{for}\:{the}\:{derivitive}\:{of} \\ $$$${the}\:{gamma}\:{function}. \\ $$
Commented by tawa tawa last updated on 09/Jul/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$