Question

Вычислить производную, №1

Answer 1

$1)~x^x=e^{\ln x^x}=e^{x\ln x}\\\\~~\Big(x^x\Big)'=\Big(e^{x\ln x}\Big)'=e^{x\ln x} \big(x\ln x\big)'=\\\\~~=e^{\ln x^x}\big(x'\cdot \ln x+x\cdot (\ln x)'\big)=x^x\bigg(\ln x+\dfrac xx\bigg)\\\\~~\boxed{\boldsymbol{\Big(x^x\Big)'=x^x\big(\ln x+1\big)}}$

$2)~x^{x^x}=e^{\ln x^{x^x}}=e^{x^x\ln x}\\\\\Big(x^{x^x}\Big)'=\Big(e^{x^x\ln x}\Big)'=e^{x^x\ln x} \big(x^x\ln x\big)'=\\\\~~~=e^{\ln x^{x^x}}\Big(\big(x^x\big)'\cdot \ln x+x^x\cdot (\ln x)'\big)=\\\\~~~=x^{x^x}\bigg(x^x\Big(\ln x+1\Big)\ln x+\dfrac {x^x}x\bigg)=\\\\~~~=x^{x^x}\bigg(x^x\Big(\ln^2 x+\ln x\Big)+x^{x-1}\bigg)=\\\\~~~=x^{x^x}\cdot x^x\Big(\ln^2 x+\ln x\Big)+x^{x^x+x-1}=\\\\~~~=x^{x^x+x}\Big(\ln^2 x+\ln x\Big)+x^{x^x+x-1}$

$\boldsymbol{\Big(x^x+x^{x^x}\Big)'=}\Big(x^x\Big)'+\Big(x^{x^x}\Big)'=\\\\\boldsymbol{=x^x\big(\ln x+1\big)+x^{x^x+x}\Big(\ln^2 x+\ln x\Big)+x^{x^x+x-1}}$