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    Dodo333's Avatar
    Dodo333 Posts: 3, Reputation: 1
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    #1

    May 4, 2013, 08:03 AM
    Differentiate: y=(x)^((x^x)) using dy/dx
    Please show the steps and the final answer. Thank you!
    CWH68's Avatar
    CWH68 Posts: 26, Reputation: 2
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    #2

    Apr 11, 2015, 01:04 AM
    ln(x^x) = x ln x, so x^x = e^(x ln x). Now, ln (x^(x^x)) = (x^x) ln x
    = (ln x)(e^(x ln x)). Thus, x^(x^x) = e^((ln x)(e^(x ln x))).
    Thus, dy/dx = e^((ln x)(e^(x ln x)))*d/dx((ln x)(e^(x ln x)))
    =(x^(x^x))*d/dx((ln x)(e^(x ln x))) = (x^(x^x))*((e^(x ln x))/x +
    (ln x)*(d/dx(e^x ln x))) = (x^(x^x))*((e^(x ln x))/x +
    (ln x)(e^(x ln x))(1 + ln x)) = (x^(x^x))*((x^x)/x +
    (x^x)(ln x)(1+ln x)) = (x^(x^x))(x^x)((1/x) + ln x + (ln x)^2).
    Hope this helps.
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