Why is the answer Wolfram|Alpha Widgets: "Derivative Calculator" - Free Mathematics Widget and not [e^x] [ x^(e^x -1 )]

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- Jun 12th 2014, 03:58 PMsakonpure6Derivative of x^ (e^x)
Why is the answer Wolfram|Alpha Widgets: "Derivative Calculator" - Free Mathematics Widget and not [e^x] [ x^(e^x -1 )]

- Jun 12th 2014, 04:39 PMromsekRe: Derivative of x^ (e^x)
$x^{e^x}=e^{\ln(x)e^x}$

$\dfrac d {dx} e^{\ln(x)e^x}=e^{\ln(x)e^x}\dfrac d {dx} \left(\ln(x)e^x\right)=$

$e^{\ln(x)e^x}\left(\dfrac {e^x}{x} + \ln(x)e^x\right)=x^{e^x}\left(\dfrac {e^x}{x} + \ln(x)e^x\right)$ - Jun 12th 2014, 05:06 PMsakonpure6Re: Derivative of x^ (e^x)
So we can't take the derivative using the power rule and chain rule for x^(e^x) without re writing it?

- Jun 12th 2014, 05:12 PMromsekRe: Derivative of x^ (e^x)
- Jun 12th 2014, 05:54 PMsakonpure6Re: Derivative of x^ (e^x)
Thank you!

- Jun 12th 2014, 06:27 PMSlipEternalRe: Derivative of x^ (e^x)
The power rule can only be used when the exponent of $x$ is a real number. Since you have the exponent of $x$ is a function of $x$, the power rule doesn't work. Romsek's method works well, as does implicit differentiation:

$y = x^{e^x}$

$\ln y = \ln x^{e^x} = e^x\ln x$

$\dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{e^x}{x}+e^x\ln x$

$\dfrac{dy}{dx} = y\left(\dfrac{e^x}{x}+e^x\ln x\right)$

Plugging in for $y = x^{e^x}$ gives:

$\dfrac{dy}{dx} = x^{e^x}\left(\dfrac{e^x}{x}+e^x\ln x\right)$, just as romsek computed. - Jun 12th 2014, 06:31 PMsakonpure6Re: Derivative of x^ (e^x)
Great, thanks for the clarification Sip