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Problem in the derivate of x^x

  1. Mar 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Why is the derivate of [tex]x^x[/tex]
    [tex]x^x(1 + lnx)[/tex]?

    3. The attempt at a solution
    I know the derivate can be explained by calculus.
    However, I am not exactly sure how.
  2. jcsd
  3. Mar 4, 2009 #2
    One way to approach the problem is from the equivalence


    Which gives that




    This you should be able to derivate using the set of rules you already know.
  4. Mar 4, 2009 #3


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    Science Advisor

    Or, let y= xx. Then ln(y)= x ln (x). Differentiate both sides with respect to x (implicit differentiation) and solve for dy/dx.
  5. Mar 4, 2009 #4
    Thank you!

    So we get the solution by differenting
    [tex]y = e^{g(x)}[/tex],
    where [tex]g(x) = xlnx[/tex].

    [tex] y' = g'(x) * e^{g(x)}[/tex],
    where g'(x) = lnx + 1
    [tex] y' = (lnx + 1) * e^{g(x)}[/tex]
    [tex] y' = (lnx + 1) x^{x}[/tex]
  6. Mar 4, 2009 #5
    Thank you HallofIvy!
    Your answer is excellent.
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