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

 

 

 

 
 
 
 
 

Answers (1)

y= \left ( x \right )^{\cos x}+\left ( \cos x \right )^{\sin x}
\Rightarrow y= e^{\log_{e}\left ( x \right )^{\cos x}}+e^{\log_{e}\left ( \cos x \right )^{\sin x}}
\Rightarrow y=e^{\cos x\, \log_{e}x}+e^{\sin x\, \log_{e}\left ( \cos x \right )}
\therefore \frac{dy}{dx}= e^{\cos x\log_{e}}\, ^{x}\left [ \cos x\times \frac{1}{x}- \log_e x \sin x \right ]
                + e^{\sin x \log_{e}\cos x}\left [ \sin x \frac{-\sin x}{\cos x}+\log \left ( \cos x \right )\times \left ( \cos x \right ) \right ]
That is, \frac{dy}{dx}= x^{\cos x}\left ( \frac{\cos x}{x}-\sin x\, \log x \right )+\left ( \cos x \right )^{\sin x}\left ( -\sin x\tan x+\cos x\log \left ( \cos x \right ) \right ) 


 

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