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Question Number 142045 by Engr_Jidda last updated on 25/May/21

Answered by Dwaipayan Shikari last updated on 25/May/21

$$y={\left(\frac{1}{x}\right)}^x$$$$log\left(y\right)=-xlog\left(x\right)\Rightarrow \frac{y^{\prime }}{y}=-1-log\left(x\right)\Rightarrow y=-x^{-x}(1+log\left(x\right))$$

Commented by Engr_Jidda last updated on 25/May/21

$$thanks$$

Answered by EDWIN88 last updated on 26/May/21

$$\left(1\right)\mathrm{y}=\log \left(\cos \mathrm{x}\right)\Rightarrow \cos \mathrm{x}={\mathrm{e}}^{\mathrm{y}}$$$$\Rightarrow -\sin \mathrm{x}={\mathrm{e}}^{\mathrm{y}}.{\mathrm{y}}^{\prime }$$$$\Rightarrow {\mathrm{y}}^{\prime }=-\sin \mathrm{x}.{\mathrm{e}}^{-\mathrm{y}}=-\sin \mathrm{x}.{\mathrm{e}}^{-\log \left(\cos \mathrm{x}\right)}$$$$\Rightarrow {\mathrm{y}}^{\prime }=-\sin \mathrm{x}.\sec \mathrm{x}$$

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