Derivative of (sin(x))^x

  • Thread starter The_Brain
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  • #1
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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.
 

Answers and Replies

  • #2
Edit:

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

Now differentiate implicitly.

cookiemonster
 
Last edited:
  • #3
23
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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.
 
  • #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.
 
  • #5
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dw soz
 
Last edited:
  • #6
83
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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:
  • #7
def
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uh guys, are you even allowed to do ln(sinx), i mean sinx isn't always positive right?
 
  • #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.
 
  • #9
def
4
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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)?
 
  • #10
def
4
0
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
 
  • #11
HallsofIvy
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The problem is not just differentiating [itex](-1)^x[/itex]. How are you defining [itex](-1)^x[/itex]?
 
  • #12
def
4
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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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