derivative of x^{cosx}

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ddx (x^cos(x))=x^cos(x)(−sin(x)ln(x)+cos(x)x )

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ddx (x^cos(x))=x^cos(x)(−sin(x)ln(x)+cos(x)x )

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ddx (x^cos(x))

Apply exponent rule: a^b=e^bln(a)
x^cos(x)=e^cos(x)ln(x)
=ddx (e^cos(x)ln(x))

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Apply the chain rule: e^cos(x)ln(x)ddx (cos(x)ln(x))
=e^cos(x)ln(x)ddx (cos(x)ln(x))

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ddx (cos(x)ln(x))=−sin(x)ln(x)+cos(x)x
=e^cos(x)ln(x)(−sin(x)ln(x)+cos(x)x )

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e^cos(x)ln(x)=x^cos(x)
=x^cos(x)(−sin(x)ln(x)+cos(x)x )