Question Number 81439 by jagoll last updated on 13/Feb/20
$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{\mathrm{x}} \\ $$$$\mathrm{f}^{\left(\mathrm{100}\right)} \left(\mathrm{0}\right)\:=\:? \\ $$
Commented by john santu last updated on 13/Feb/20
$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{\mathrm{x}} \:=\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{x}^{\mathrm{n}+\mathrm{2}} }{\mathrm{n}!\:} \\ $$$$\Rightarrow\mathrm{n}+\mathrm{2}\:=\:\mathrm{100}\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\mathrm{98}} {\sum}}\:\frac{\mathrm{x}^{\mathrm{100}} }{\mathrm{98}!} \\ $$$$\mathrm{f}\:^{\left(\mathrm{100}\right)} \:\left(\mathrm{0}\right)\:=\:\frac{\mathrm{100}!}{\mathrm{98}!}\:=\:\mathrm{100}×\mathrm{99}\:=\:\mathrm{9900} \\ $$
Commented by abdomathmax last updated on 13/Feb/20
$${f}^{\left({n}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({x}^{\mathrm{2}} \right)^{\left({k}\right)} \:\left({e}^{{x}} \right)^{\left({n}−{k}\right)} \\ $$$$={C}_{{n}} ^{\mathrm{0}} {x}^{\mathrm{2}} {e}^{{x}} \:+\mathrm{2}{x}\:{C}_{{n}} ^{\mathrm{1}} \:{e}^{{x}} \:+\mathrm{2}\:{C}_{{n}} ^{\mathrm{2}} \:{e}^{{x}} \\ $$$$={x}^{\mathrm{2}} \:{e}^{{x}} \:+\mathrm{2}{nx}\:{e}^{{x}} \:+\mathrm{2}\:×\frac{{n}!}{\mathrm{2}!\left({n}−\mathrm{2}\right)!}\:{e}^{{x}} \\ $$$$=\left({x}^{\mathrm{2}} \:+\mathrm{2}{nx}\:\:+{n}\left({n}−\mathrm{1}\right)\right){e}^{{x}} \:\Rightarrow \\ $$$${f}^{\left(\mathrm{100}\right)} \left(\mathrm{0}\right)\:=\mathrm{100}×\mathrm{99}=\mathrm{9900} \\ $$