The binary operation is only defined for .
For example, for is defined
.
You can differenciate it taking previously logarithms:
and now, differenciate the equality.
Regards.
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Fernando Revilla
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!!
The binary operation is only defined for .
For example, for is defined
.
You can differenciate it taking previously logarithms:
and now, differenciate the equality.
Regards.
---
Fernando Revilla
I understand. Perhaps you wanted to lead the question out of context.
(i) .
(ii) .
(iii) is well defined if .
(iv) does not exists in .
(v) (contradiction with (iv).
(vi) 0,\pi)\rightarrow{\mathbb{R}},\;f(x)=(\sin x)^x" alt="f0,\pi)\rightarrow{\mathbb{R}},\;f(x)=(\sin x)^x" /> is well defined.
etc, etc, etc, ...
Regards.
---
Fernando Revilla
Well, of course the expression is ambiguous.
I interpreted .
Regards.
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Fernando Revilla
Just what you have written is not ambiguous, the notation is ambiguous, evidently ambiguous.
Regards.
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Fernando Revilla
No problem, it was a lapse of concentration on my part to write instead of .
Regards.
---
Fernando Revilla
What You say is perfectly correct and is...
(1)
Because the exponential function is defined for all real or complex values of the exponent, the expression (1) is perfectly computable. The for 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
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