1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

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

    Now differentiate implicitly.

    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
    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

    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


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


    User Avatar

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


    User Avatar
    Staff Emeritus
    Science Advisor

    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


    User Avatar

    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


    User Avatar

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

    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


    User Avatar
    Staff Emeritus
    Science Advisor

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


    User Avatar

    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 :(
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Derivative of (sin(x))^x
  1. X->0 (sin[x])/x (Replies: 3)

  2. Derivative of x^sin(x) (Replies: 9)

  3. Integral sin(x)/x (Replies: 2)