Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 145338 by puissant last updated on 04/Jul/21

$$\mathrm{d}\acute{\mathrm{e}riv}\acute{\mathrm{e}e}\mathrm{n}-\mathrm{i}\stackrel{`}{\mathrm{e}me}\mathrm{de}\frac{{\mathrm{x}}^3}{1+{\mathrm{x}}^6}$$

Answered by mathmax by abdo last updated on 04/Jul/21

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^3}{1+{\mathrm{x}}^6}$$$${\mathrm{x}}^6+1=0\Rightarrow {\mathrm{x}}^6={\mathrm{e}}^{\mathrm{i}(2\mathrm{k}+1)\pi }\Rightarrow \mathrm{roots}\mathrm{are}{\mathrm{x}}_{\mathrm{k}}={\mathrm{e}}^{\mathrm{i}(2\mathrm{k}+1)\frac{\pi }{6}}$$$$\mathrm{k}\in \left[[0,5]\right]\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^3}{\prod _{\mathrm{k}=0}^5(\mathrm{x}-{\mathrm{x}}_{\mathrm{k}})}=\sum _{\mathrm{k}=0}^5\frac{{\mathrm{a}}_{\mathrm{k}}}{\mathrm{x}-{\mathrm{x}}_{\mathrm{k}}}$$$${\mathrm{a}}_{\mathrm{k}}=\frac{{\mathrm{x}}_{\mathrm{k}}^3}{{6\mathrm{z}}_{\mathrm{k}}^5}=-\frac{{\mathrm{x}}_{\mathrm{k}}^4}{6}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=-\frac{1}{6}\sum _{\mathrm{k}=0}^5\frac{{\mathrm{x}}_{\mathrm{k}}^4}{\mathrm{x}-{\mathrm{x}}_{\mathrm{k}}}$$$$=-\frac{1}{6}\sum _{\mathrm{k}=0}^5\frac{{\mathrm{e}}^{\mathrm{i}(2\mathrm{k}+1)\frac{2\pi }{3}}}{\mathrm{x}-{\mathrm{x}}_{\mathrm{k}}}\Rightarrow {\mathrm{f}}^{\left(\mathrm{n}\right)}\left(\mathrm{x}\right)=-\frac{1}{6}\sum _{\mathrm{k}=0}^5{\mathrm{e}}^{\mathrm{i}(2\mathrm{k}+1)\frac{2\pi }{3}}\times \frac{{(-1)}^{\mathrm{n}}\mathrm{n}!}{{(\mathrm{x}-{\mathrm{x}}_{\mathrm{k}})}^{\mathrm{n}+1}}$$$$\Rightarrow {\mathrm{f}}^{\left(\mathrm{n}\right)}\left(\mathrm{x}\right)=\frac{{(-1)}^{\mathrm{n}+1}\mathrm{n}!}{6}\sum _{\mathrm{k}=0}^5\frac{{\mathrm{e}}^{\mathrm{i}(2\mathrm{k}+1)\frac{2\pi }{3}}}{{(\mathrm{x}-{\mathrm{x}}_{\mathrm{k}})}^{\mathrm{n}+1}}$$

Commented by puissant last updated on 04/Jul/21

$$\mathrm{Thanks}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com