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Question Number 11547 by Nayon last updated on 28/Mar/17

$$\frac{d}{dx}\left(x^x\right)=?\left[pleasegivetheanswerwithproof\right]$$

Answered by sma3l2996 last updated on 28/Mar/17

$$x^x=e^{xln\left(x\right)}$$$$so\frac{d\left(x^x\right)}{dx}=\frac{d\left(e^{xln\left(x\right)}\right)}{dx}=\frac{d}{dx}\left(xln\left(x\right)\right)e^{ln\left(x^x\right)}$$$$=(ln\left(x\right)+1)x^x$$

Answered by ajfour last updated on 28/Mar/17

$$y=x^x$$$$\ln y=x\ln x$$$$\frac{1}{y}\frac{dy}{dx}=\ln x+1$$$$\frac{dy}{dx}=y(1+\ln x)$$$$\frac{dy}{dx}=x^x(1+\ln x).$$

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