If y= \left ( \log x \right )^{x}+x^{\log x},  find \frac{dy}{dx}.

 

 

 

 
 
 
 
 

Answers (1)

y= \left ( \log x \right )^{x}+x^{\log x},
\Rightarrow y=e^{\log\left ( \log x \right )^{x}}+e^{\log x^{ \log x}}
\Rightarrow y=e^{x\log \left ( \log x \right )}+e^{\log x\, \log x}
\Rightarrow \frac{dy}{dx}= e^{x\log \left ( \log x \right )}\left \{ x\times \frac{1}{\log x}\times \frac{1}{x}+\log \left ( \log x \right )\cdot 1 \right \}
               +e^{\log x\log x}\left \{ \log x\frac{1}{x} +\log x\times \frac{1}{x}\right \}
\Rightarrow \frac{dy}{dx}= \left ( \log x \right )^{x}\left \{ \frac{1}{\log x}+\log \left ( \log x \right ) \right \}+x^{\log x}\left \{ \frac{2\log x}{x} \right \}

i.e \Rightarrow \frac{dy}{dx}= \left ( \log x \right )^{x}\left \{ \frac{1}{\log x}+\log \left ( \log x \right ) \right \}+2\, x^{\left ( \log x-1 \right )}\log x

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