# Derivative of (sin(x))^x

1. Apr 5, 2004

### The_Brain

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. Apr 5, 2004

Edit:

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

Now differentiate implicitly.

Last edited: Apr 5, 2004
3. Apr 6, 2004

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. Oct 7, 2010

### insipidgirl

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. Oct 7, 2010

dw soz

Last edited: Oct 7, 2010
6. Oct 14, 2010

### darkside00

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
7. Nov 10, 2010

### def

uh guys, are you even allowed to do ln(sinx), i mean sinx isn't always positive right?

8. Nov 10, 2010

### HallsofIvy

Staff Emeritus
Then do it for $0\le x\le \pi$ and use $sin(x+ n\pi)= (-1)^n sin(x)$ for x outside that range.

9. Nov 10, 2010

### def

kk, so how do i differentiate (-1)^x?

my question is: how do do ln((-1)^n*sinx)?

10. Nov 10, 2010

### def

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. Nov 11, 2010

### HallsofIvy

Staff Emeritus
The problem is not just differentiating $(-1)^x$. How are you defining $(-1)^x$?

12. Nov 11, 2010

### def

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 :(