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Thread: What is the derivative of $x^{x^x}$?

  1. #1
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    What is the derivative of $x^{x^x}$?

    What is the derivative of $x^{x^x}$?
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  2. #2
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    Re: What is the derivative of $x^{x^x}$?

    $y=x^{x^x}$

    We need to find derivative of $y $with respect to $x.$ So solve it using log-

    $ln\,y = x^x\,\ln\,x$

    Now, use the product formula

    $\dfrac{d}{dx}(f(x)g(x))=f(x)\dfrac{d}{dx}g(x)+g(x )\dfrac{d}{dx}f(x)$

    Here,

    $f(x)=x^x$

    $g(x)=\ln\,x$

    Substitute values in formula-

    $\dfrac{1}{y}\dfrac{dy}{dx}=(x^x)\dfrac{d}{dx} \ln\,x+\ln\,x\dfrac{d}{dx}x^x $

    $\dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{1}{x}x^x+x^x( \ln \,x+1)\ln\,x $

    $\dfrac{dy}{dx}=y(x^{x-1}+x^x(\ln\,x+1)\ln\,x) $

    $\dfrac{dy}{dx}=x^{x^x}(x^{x-1}+x^x(\ln\,x+1)\ln\,x) $
    Last edited by deesuwalka; Jan 7th 2017 at 02:46 AM.
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