Derivative of this exponential function?

**mneox** · April 7th 2010, 06:17 PM

what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

**harish21** · April 7th 2010, 06:32 PM

Originally Posted by mneox

what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

Follow this link and click on "Show steps":

differentiate x^(e^x) - Wolfram|Alpha

**tonio** · April 7th 2010, 06:33 PM

Originally Posted by mneox

what is the derivative of x^(e^x)?

Is it [e^x][x^(e^x - 1)]?

Or is there something else that I have to do. thanks!

Write $f(x):=x^{e^x}=e^{e^x\ln x}\Longrightarrow f'(x)=e^{e^x\ln x}\left[e^x\ln x+\frac{e^x}{x}\right]$ $=x^{e^x}\left[e^x\ln x+\frac{e^x}{x}\right]$ $=x^{e^x}\ln\left(x^{e^x}\right)+e^xx^{e^x-1}$ .

Tonio

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