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Question Number 81226 by 20092104 last updated on 10/Feb/20
(d/dx)(x!) and (d/dx)(1+2+3+...+x)
$$\frac{{d}}{{dx}}\left({x}!\right)\:{and}\:\frac{{d}}{{dx}}\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+{x}\right) \\ $$
Answered by MJS last updated on 10/Feb/20
(d/dx)[x!] doesn′t exist because x! is not continuous  1+2+3+...+x=((x(x+1))/2)  (d/dx)[((x(x+1))/2)]=x+(1/2)
$$\frac{{d}}{{dx}}\left[{x}!\right]\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist}\:\mathrm{because}\:{x}!\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous} \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{3}+…+{x}=\frac{{x}\left({x}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\frac{{d}}{{dx}}\left[\frac{{x}\left({x}+\mathrm{1}\right)}{\mathrm{2}}\right]={x}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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