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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

    e^(lnx)=x

    Which gives that

    e^(lnx^x)=x^x

    Or

    e^(xlnx)=x^x

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

    HallsofIvy

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    Staff Emeritus
    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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