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Derivative of (sin(x))^x

  1. Apr 5, 2004 #1
    What are the steps used to obtain the derivative of (sin(x))^x? I know it's (sin(x))^x [xcot(x) + ln(sinx)] however I don't know how to get there.
     
  2. jcsd
  3. Apr 5, 2004 #2
    Edit:

    y = sin(x)^x
    ln(y) = x ln(sin(x))

    Now differentiate implicitly.

    cookiemonster
     
    Last edited: Apr 5, 2004
  4. Apr 6, 2004 #3
    or: (d/dx)sin(x)^x = (d/dx)e^xln(sin(x)) = (ln(sin(x))+xcosx/sinx)sin(x)^x, that's what you got.
     
  5. Oct 7, 2010 #4
    1st
    lny=ln((sinx)^x)
    lny=xln(sinx)
    then take the derivative and use product and chain rule ;)

    dy/dx(1/y)=x(1/sinx)cosx + ln(sinx)
    then simplfy

    dy/dx(1/y) = xcotx+lnsinx
    dy/dx=y[xcotx+lnsinx]

    plug in for y

    dy/dx= (sinx)^x[xcotx+lnsinx]

    message me if you need any explanations.
     
  6. Oct 7, 2010 #5
    dw soz
     
    Last edited: Oct 7, 2010
  7. Oct 14, 2010 #6
    Thats a fun one to solve.

    how do you integrate (sin(x))^x?

    For the second derivative, I got

    (((sin(x))^x)*(ln(sinx)+x/tanx))*(ln(sinx)+x*cotx)+((sin(x))^x)*(cotx-cotx/((sin(x))^2)

    but got no way of checking
     
    Last edited: Oct 15, 2010
  8. Nov 10, 2010 #7

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    uh guys, are you even allowed to do ln(sinx), i mean sinx isn't always positive right?
     
  9. Nov 10, 2010 #8

    HallsofIvy

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    Then do it for [itex]0\le x\le \pi[/itex] and use [itex]sin(x+ n\pi)= (-1)^n sin(x)[/itex] for x outside that range.
     
  10. Nov 10, 2010 #9

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    kk, so how do i differentiate (-1)^x?
    edit: sorry i misread your post

    my question is: how do do ln((-1)^n*sinx)?
     
  11. Nov 10, 2010 #10

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    if I take -Pi<x<0 then I can write:

    sin(x)^x=(-1)^x*(-sin(x))^x
    so sin(x)^x=(-1)^x*exp(x*ln(-sin(x))) but then I'm stuck in the differentiation because of the (-1)^x
     
  12. Nov 11, 2010 #11

    HallsofIvy

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    The problem is not just differentiating [itex](-1)^x[/itex]. How are you defining [itex](-1)^x[/itex]?
     
  13. Nov 11, 2010 #12

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    I dunno, hence my question: how do you guys manage to differentiate sinx^x?

    When I try to differentiate it using the suggested method which I get stuck because of the (-1)^x for values of x on the intervals of the form [2kPi-Pi,2kPi].

    This is frustrating I know, everywhere I look people use the same method, but to me there is something missing , or maybe there is something wrong with my thinking :(
     
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