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Math Help - the derivative of y=x^x^x

  1. #1
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    the derivative of y=x^x^x

    Find derivative of y=x^x^x.

    Whoa. Not sure how to attack this but heres what i think:

    use logarithmic diff, so it would be xln(x^x)
    Then product rule, so x*derivative of ln(x^x)+ln(x^x), or can i do the same law of logs again to make it x*derivative of ln(xln(x))+ln(x)?
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  2. #2
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    Quote Originally Posted by Evan.Kimia View Post
    Find derivative of y=x^x^x.

    Whoa. Not sure how to attack this but heres what i think:

    use logarithmic diff, so it would be xln(x^x)
    Then product rule, so x*derivative of ln(x^x)+ln(x^x), or can i do the same law of logs again to make it x*derivative of ln(xln(x))+ln(x)?

    First, You should specify it:
    Is it y=(x^x)^x or y=x^{(x^x)} ?

    Surely, for both cases you will use the logarithmic differentiation.
    But there is a big difference between them when you do the Log.D.
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  3. #3
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    its y=x^(x^x). I was about to edit this post: Can i just make it x^(x^2)?
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    Quote Originally Posted by Evan.Kimia View Post
    its y=x^(x^x). I was about to edit this post: Can i just make it x^(x^2)?
     <br />
x\times x = x^2 \neq x^x<br />
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  5. #5
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    Quote Originally Posted by Evan.Kimia View Post
    its y=x^(x^x). I was about to edit this post: Can i just make it x^(x^2)?
    No !
    y=(x^x)^x=x^{ (x^2) }
    But y=x^{(x^x)} \neq x^{ x^2 }
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  6. #6
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    Be careful with your notation! y= (x^x)^x= x^{x^2} but x^{(x^x)}\ne x^{x^2}.

    If y= x^{(x^x)}, then ln(y)= x^x ln(x) and, taking the logarithm again, ln(ln(y))= x ln(x)+ ln(x).
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  7. #7
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    Quote Originally Posted by HallsofIvy View Post
    Be careful with your notation! y= (x^x)^x= x^{x^2} but x^{(x^x)}\ne x^{x^2}.

    If y= x^{(x^x)}, then ln(y)= x^x ln(x) and, taking the logarithm again, ln(ln(y))= x ln(x)+ {\color{red}ln(x)}.
    It should be ln(ln(x)).
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  8. #8
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    I figure it out. I tried applying ln again, but it got wayy to sloppy. Since i solved the derivative for x^x before separately while trying to do this problem i was able to just plug in the derivative when doing the product rule on ln(y)=x^xln(x). My final answer was x^x^x[((x^x)*1/x)+(ln(x)*x^x(ln(x)+1)]. I then found this video on youtube and the guy did exactly what i did . Thanks for all the help!

    If you want to check out the video, its at
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