1. ## Help differentiating sin(x)^x

Hi,

How do you differentiate sin(x)^x for all real numbers? How do you even define sin(x)^x for all real numbers?

I'm having problems doing this rigorously. Sure you can use the fact that sin(x)^x = exp(x*ln(sin(x))), but what happens when x belongs to an interval of the form [2*k*Pi-Pi,2*k*Pi]?

I'm doing first year math and our lecturer gave this function to differentiate. Is it me or is this much too hard? My graphic calculator won't even trace it let alone differentiate it!!

2. Originally Posted by def
Hi,
How do you differentiate sin(x)^x for all real numbers? How do you even define sin(x)^x for all real numbers?
The binary operation $\displaystyle a^b$ is only defined for $\displaystyle a>0$.

For example, for $\displaystyle x\in (0,\pi)$ is defined

$\displaystyle f(x)=(\sin x)^x$.

You can differenciate it taking previously logarithms:

$\displaystyle \ln f(x)=x\ln (\sin x)$

and now, differenciate the equality.

Regards.

---
Fernando Revilla

3. def, does your assigned problem actually state that $\displaystyle \left(\sin x\right)^x$ is defined for all real numbers? Because it sounds like you are really just expected to take the derivative without worrying about the domain.

4. Originally Posted by drumist
This is of course not true.
I understand. Perhaps you wanted to lead the question out of context.

(i) $\displaystyle (-2)^3=-8$.

(ii) $\displaystyle (-1)^{i}=e^{i\log(-1)}=e^{i(\pi i)}=e^{-\pi}$.

(iii) $\displaystyle f:\mathbb{R}\rightarrow{\mathbb{R}},\;f(x)=a^x$ is well defined if $\displaystyle a>0$.

(iv) $\displaystyle (-1)^{1/2}=\sqrt{-1}$ does not exists in $\displaystyle \mathbb{R}$.

(v) $\displaystyle (-1)^{1/2}=(-1)^{2/4}=\sqrt[4]{(-1)^2}= \pm 1$ (contradiction with (iv).

(vi) $\displaystyle f0,\pi)\rightarrow{\mathbb{R}},\;f(x)=(\sin x)^x$ is well defined.

etc, etc, etc, ...

Regards.

---
Fernando Revilla

5. Its $\displaystyle sin^x(x)$ or $\displaystyle sin\left(x^x\right)$ ?

6. Originally Posted by General
Its $\displaystyle sin^x(x)$ or $\displaystyle sin\left(x^x\right)$ ?
Well, of course the expression $\displaystyle \sin (x)^x$ is ambiguous.

I interpreted $\displaystyle (\sin x)^x$.

Regards.

---
Fernando Revilla

7. Nope.
Its not ambiguous.

$\displaystyle \dfrac{d}{dx}\left( sin\left(x^x\right)\right)= cos\left(x^x\right) \cdot \dfrac{d}{dx}\left(x^x\right)$

8. Originally Posted by General
Nope.
Its not ambiguous.

$\displaystyle \dfrac{d}{dx}\left( sin\left(x^x\right)\right)= cos\left(x^x\right) \cdot \dfrac{d}{dx}\left(x^x\right)$
Just what you have written is not ambiguous, the notation $\displaystyle \sin (x)^x$ is ambiguous, evidently ambiguous.

Regards.

---
Fernando Revilla

9. Originally Posted by FernandoRevilla
I understand. Perhaps you wanted to lead the question out of context.
I said it poorly, but I meant to suggest that the domain of the given function does include some negative numbers. (The mistake you now corrected wrt the domain also confused me.)

10. Originally Posted by drumist
I said it poorly, but I meant to suggest that the domain of the given function does include some negative numbers. (The mistake you now corrected wrt the domain also confused me.)
No problem, it was a lapse of concentration on my part to write $\displaystyle (-\pi,\pi)$ instead of $\displaystyle (0,\pi)$.

Regards.

---
Fernando Revilla

11. Originally Posted by def
Hi,

How do you differentiate sin(x)^x for all real numbers? How do you even define sin(x)^x for all real numbers?

I'm having problems doing this rigorously. Sure you can use the fact that sin(x)^x = exp(x*ln(sin(x))), but what happens when x belongs to an interval of the form [2*k*Pi-Pi,2*k*Pi]?

I'm doing first year math and our lecturer gave this function to differentiate. Is it me or is this much too hard? My graphic calculator won't even trace it let alone differentiate it!!
What You say is perfectly correct and is...

$\displaystyle f(x)= (\sin x)^{x} = e^{x\ \ln \sin x}$ (1)

Because the exponential function is defined for all real or complex values of the exponent, the expression (1) is perfectly computable. The $\displaystyle f(x)$ for $\displaystyle -2\ \pi< x < 2\ \pi$ is represented here...

Of course the function is 'a little demential' ... it is composed by a real and imaginary part [represented in red and blue in the figure...] and, in my opinion, is of little interest... exactly as its derivative ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

12. Thanks for all your helpful comments. Sorry for the ambiguous notation. I did indeed mean [sin(x)]^x.

The question itself was not clear as the lecturer asked us to "differentiate the function for real values of x".

Of course there will be an imaginary part if sin(x) is negative, but I doubt that he expects us to study the function for these values of x.

Regards