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Question Number 199176 by cortano12 last updated on 29/Oct/23

$$\mathrm{If}\mathrm{f}\left(\mathrm{x}\right)=({\mathrm{x}}^2-4\mathrm{x})\sin 4\mathrm{x}$$$$\mathrm{find}{\mathrm{f}}^{\left(6\right)}\left(\mathrm{x}\right).$$

Answered by MM42 last updated on 29/Oct/23

$$f=u.v$$$$\Rightarrow f^{\left(n\right)}=\underset{i=0}{\sum ^n}\left(\begin{array}{c}n\\ i\end{array}\right)\times u^{(n-i)}\times v^{\left(i\right)}$$$$let:u=x^2-4x\mathrm{\&}v=sin4x$$$$\Rightarrow f^{\left(6\right)}\left(x\right)=30\times 4^{4}sin4x+6\times 4^5\times (2x-4)cos4x-4^6\times (x^2-4x)sin4x✓$$

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