Question Number 196299 by hardmath last updated on 22/Aug/23
$$\mathrm{If}\:\rightarrow\:\mathrm{y}\:=\:\mathrm{x}\:!\:\:\:\:\:\mathrm{find}\:\rightarrow\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$
Answered by Frix last updated on 22/Aug/23
$${y}={x}!\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{N}\:\Rightarrow\:\mathrm{no}\:\mathrm{derivative} \\ $$$$\mathrm{exists}. \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{mean}\:{x}!=\Gamma\:\left({x}+\mathrm{1}\right): \\ $$$${y}=\Gamma\:\left({x}+\mathrm{1}\right)\:=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{t}^{{x}} }{\mathrm{e}^{{t}} }\:{dt}\:\Rightarrow\:\frac{{dy}}{{dx}}=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{t}^{{x}} \mathrm{ln}\:{t}}{\mathrm{e}^{{t}} }\:{dt} \\ $$