[(cos(x))^x]'

I'm supposed to find the derivative of [cos(x)]^x using logarithmic differentiation. Please help.
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Let y=cosx(x), ln(y)=xln(cos(x)).

(1/y)y'=ln(cos(x))+x(1/cos(x))(-sin(x))=ln(cos(x))-xtan(x),

y'=y(ln(cos(x))-xtan(x))=cosx(x)(ln(cos(x))-xtan(x)).

So the derivative is cosx(x)(ln(cos(x))-xtan(x)).

ago by Top Rated User (1.2m points)

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