Calcule a derivada de y = (sen x)^(e^x) ?

-> y = (sen x)^(e^x)

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Usando a propriedade z = e^(ln z):

-> y = e^[ ln (sen x)^(e^x) ]

Usando a propriedade ln z^w = w·ln z:

-> y = e^[ e^x · ln (sen x) ]

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Derivando:

-> y' = { e^[ e^x · ln (sen x) ] }'

-> y' = e^[ ln (sen x)^(e^x) ] · { e^x · ln (sen x) }'

-> y' = (sen x)^(e^x) · { [ e^x ]' · ln (sen x) + e^x · [ ln (sen x) ]' }

-> y' = (sen x)^(e^x) · { e^x · ln (sen x) + e^x · ( 1/sen x ) · ( sen x )' }

-> y' = (sen x)^(e^x) · e^x ·{ ln (sen x) + ( 1/sen x ) · ( cos x ) }

-> y' = (sen x)^(e^x) · e^x · { ln (sen x) + cotg x }

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