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If \mathrm{f(x)=\cot ^{-1}\left(\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right)+\cot ^{-1}\left(\frac{\log _e\left(e x^4\right)}{\log _e\left(e^2 / x^2\right)}\right) then\, \, f^n(1) } equals, where \mathrm{ f^n(x) \, denotes\, \, the\, \, n^{\text {th }} } order derivative of f(x)

Option: 1

1


Option: 2

0


Option: 3

3


Option: 4

-1


Answers (1)

best_answer

\mathrm{\begin{aligned} & \text { } f(x)=\cot ^{-1}\left(\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right)+\cot ^{-1}\left(\frac{\log _e\left(e x^4\right)}{\log _e\left(\frac{e^2}{x^2}\right)}\right) \\ & =\tan ^{-1}\left(\frac{1+\log x^2}{1-\log x^2}\right)+\tan ^{-1}\left(\frac{2-\log x^2}{1+2 \log x^2}\right) \\ & =\tan ^{-1} 1+\tan ^{-1}\left(\log x^2\right)+\tan ^{-1} 2-\tan ^{-1}\left(\log x^2\right) \\ & \therefore f(x)=\tan ^{-1} 1+\tan ^{-1} 2 \\ & \Rightarrow f(x)=\operatorname{constant} \text { function } \\ & \Rightarrow f^{\prime}(x)=f^{\prime \prime}(x)=\ldots . .=f^n(x)=0 \end{aligned} }

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vishal kumar

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