# Q3  Differentiate the functions w.r.t. x.   $(\log x ) ^{\cos x}$

Given function is
$y=(\log x ) ^{\cos x}$
take log on both the sides
$\log y=\cos x\log (\log x )$
Now, differentiation w.r.t x is
$\frac{d(\log y)}{dx} = \frac{d(\cos x\log(\log x))}{dx}\\ \frac{1}{y}.\frac{dy}{dx} = (-\sin x)(\log(\log x)) + \cos x.\frac{1}{\log x}.\frac{1}{x}\\ \frac{dy}{dx}= y( \cos x.\frac{1}{\log x}.\frac{1}{x}-\sin x\log(\log x) )\\ \frac{dy}{dx} = (\log x)^{\cos x}( \frac{\cos x}{x\log x}-\sin x\log(\log x) )$
Therefore, the answer is  $(\log x)^{\cos x}( \frac{\cos x}{x\log x}-\sin x\log(\log x) )$

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