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u-v-u-v-tow-fonctions-




Question Number 135536 by islamo last updated on 13/Mar/21
(u^v )′=?  u ,v   tow fonctions
$$\left(\boldsymbol{{u}}^{\boldsymbol{{v}}} \right)'=? \\ $$$${u}\:,{v}\:\:\:{tow}\:{fonctions} \\ $$
Answered by Olaf last updated on 13/Mar/21
u^v  = e^(ln(u^v ))  = e^(vlnu)   (u^v )′ = (vlnu)′e^(vlnu)  = (v′lnu+((vu′)/u))u^v   (u^v )′ = v′u^v lnu+u′vu^(v−1)
$${u}^{{v}} \:=\:{e}^{\mathrm{ln}\left({u}^{{v}} \right)} \:=\:{e}^{{v}\mathrm{ln}{u}} \\ $$$$\left({u}^{{v}} \right)'\:=\:\left({v}\mathrm{ln}{u}\right)'{e}^{{v}\mathrm{ln}{u}} \:=\:\left({v}'\mathrm{ln}{u}+\frac{{vu}'}{{u}}\right){u}^{{v}} \\ $$$$\left({u}^{{v}} \right)'\:=\:{v}'{u}^{{v}} \mathrm{ln}{u}+{u}'{vu}^{{v}−\mathrm{1}} \\ $$

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