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Question Number 27081 by abdo imad last updated on 01/Jan/18
$${let}\:{give}\:{f}\left({x}\right)=\:\:\frac{{x}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{1}}\:\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:\:. \\ $$
Answered by prakash jain last updated on 01/Jan/18
$${f}\left({x}\right)\frac{{x}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}{x}−\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}{x}+\mathrm{1}}\right) \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\left(−\mathrm{2}\right)^{{n}} {n}!}{\left(\mathrm{2}{x}−\mathrm{1}\right)^{{n}+\mathrm{1}} }+\frac{\left(−\mathrm{2}\right)^{{n}} {n}!}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{{n}+\mathrm{1}} }\right) \\ $$