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

$$y={lnx}^{x^x}{y}^{{}^{\prime }}=?$$

Answered by shunmisaki007 last updated on 22/Jul/23

$$y=\ln \left(x^{x^x}\right)=x^x\ln \left(x\right)=e^{\ln \left(x^x\right)}\ln \left(x\right)=e^{x\ln \left(x\right)}\ln \left(x\right)$$$$y^{\prime }=e^{x\ln \left(x\right)}\cdot \frac{1}{x}+e^{x\ln \left(x\right)}(x\cdot \frac{1}{x}+\ln \left(x\right))\ln \left(x\right)$$$$=e^{x\ln \left(x\right)}(\frac{1}{x}+\ln \left(x\right)+{\left(\ln \left(x\right)\right)}^2)$$$$=x^x(\frac{1}{x}+\ln \left(x\right)+{\left(\ln \left(x\right)\right)}^2)...★$$

Commented by mathlove last updated on 22/Jul/23

$$thanks$$

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