Question

Como calcular a Derivada implícita de e^x.y+y^3=1 ?

Answer 1

Lembrete inicial, para derivada de produto:

$y=ab\\\\ y' = a'.b + a.b'$



$Fun\c{c}\~ao:\\\\ e^xy+y^3=1\\\\\\\\ Resolvendo:\\\\ e^x.\dfrac{\delta x}{\delta x}.y+e^x.y^{(1-1)}.\dfrac{\delta y}{\delta x}+3y^2.\dfrac{\delta y}{\delta x}=\dfrac{\delta 1}{\delta x}\\\\\\ e^xy+e^x.\dfrac{\delta y}{\delta x}+3y^2.\dfrac{\delta y}{\delta x}=0\\\\\\ e^x.\dfrac{\delta y}{\delta x}+3y^2.\dfrac{\delta y}{\delta x}=-e^xy\\\\\\ \dfrac{\delta y}{\delta x}(e^x+3y^2)=-e^xy\\\\\\ \dfrac{\delta y}{\delta x}=\dfrac{-e^xy}{e^x+3y^2}\\\\\\ \boxed{f'_{(x)}=-\dfrac{e^xy}{e^x+3y^2}}$


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Bons estudos!