Question Number 199176 by cortano12 last updated on 29/Oct/23
$$\:\:\:\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:=\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4x}\right)\:\mathrm{sin}\:\mathrm{4x}\: \\ $$$$\:\:\:\mathrm{find}\:\mathrm{f}^{\left(\mathrm{6}\right)} \left(\mathrm{x}\right).\: \\ $$
Answered by MM42 last updated on 29/Oct/23
$${f}={u}.{v} \\ $$$$\Rightarrow{f}^{\left({n}\right)} =\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}\:\begin{pmatrix}{{n}}\\{{i}}\end{pmatrix}×{u}^{\left({n}−{i}\right)} ×{v}^{\left({i}\right)} \\ $$$${let}\::\:\:{u}={x}^{\mathrm{2}} −\mathrm{4}{x}\:\:\&\:\:{v}={sin}\mathrm{4}{x} \\ $$$$\Rightarrow{f}^{\left(\mathrm{6}\right)} \left({x}\right)=\mathrm{30}×\mathrm{4}^{\mathrm{4}} {sin}\mathrm{4}{x}+\mathrm{6}×\mathrm{4}^{\mathrm{5}} ×\left(\mathrm{2}{x}−\mathrm{4}\right){cos}\mathrm{4}{x}−\mathrm{4}^{\mathrm{6}} ×\left({x}^{\mathrm{2}} −\mathrm{4}{x}\right){sin}\mathrm{4}{x}\:\checkmark \\ $$