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Math Help - Derivative of x^ (e^x)

  1. #1
    Senior Member sakonpure6's Avatar
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    Derivative of x^ (e^x)

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

    $x^{e^x}=e^{\ln(x)e^x}$

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

    $e^{\ln(x)e^x}\left(\dfrac {e^x}{x} + \ln(x)e^x\right)=x^{e^x}\left(\dfrac {e^x}{x} + \ln(x)e^x\right)$
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    Senior Member sakonpure6's Avatar
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    Re: 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?
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    Re: Derivative of x^ (e^x)

    Quote Originally Posted by sakonpure6 View Post
    So we can't take the derivative using the power rule and chain rule for x^(e^x) without re writing it?
    no sir

    $\dfrac d {dx} x^{e^x} \neq x^{e^x - 1}$
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    Senior Member sakonpure6's Avatar
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    Re: Derivative of x^ (e^x)

    Thank you!
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  6. #6
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    Re: 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.
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    Re: Derivative of x^ (e^x)

    Great, thanks for the clarification Sip
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