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Question Number 195003 by mathlove last updated on 22/Jul/23

y=lnx^x^x           y^′ =?

$${y}={lnx}^{{x}^{{x}} } \:\:\:\:\:\:\:\:\:{y}^{'} =? \\ $$

Answered by shunmisaki007 last updated on 22/Jul/23

y=ln(x^x^x  )=x^x ln(x)=e^(ln(x^x )) ln(x)=e^(xln(x)) ln(x)  y′=e^(xln(x)) ∙(1/x)+e^(xln(x)) (x∙(1/x)+ln(x))ln(x)      =e^(xln(x)) ((1/x)+ln(x)+(ln(x))^2 )      =x^x ((1/x)+ln(x)+(ln(x))^2 ) ...★

$${y}=\mathrm{ln}\left({x}^{{x}^{{x}} } \right)={x}^{{x}} \mathrm{ln}\left({x}\right)={e}^{\mathrm{ln}\left({x}^{{x}} \right)} \mathrm{ln}\left({x}\right)={e}^{{x}\mathrm{ln}\left({x}\right)} \mathrm{ln}\left({x}\right) \\ $$$${y}'={e}^{{x}\mathrm{ln}\left({x}\right)} \centerdot\frac{\mathrm{1}}{{x}}+{e}^{{x}\mathrm{ln}\left({x}\right)} \left({x}\centerdot\frac{\mathrm{1}}{{x}}+\mathrm{ln}\left({x}\right)\right)\mathrm{ln}\left({x}\right) \\ $$$$\:\:\:\:={e}^{{x}\mathrm{ln}\left({x}\right)} \left(\frac{\mathrm{1}}{{x}}+\mathrm{ln}\left({x}\right)+\left(\mathrm{ln}\left({x}\right)\right)^{\mathrm{2}} \right) \\ $$$$\:\:\:\:={x}^{{x}} \left(\frac{\mathrm{1}}{{x}}+\mathrm{ln}\left({x}\right)+\left(\mathrm{ln}\left({x}\right)\right)^{\mathrm{2}} \right)\:...\bigstar \\ $$

Commented by mathlove last updated on 22/Jul/23

thanks

$${thanks} \\ $$

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