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Aug
22
comment Should I throw the dice again if I have rolled 4?
I don't think rolling a 4 gives you a 50% chance to win. Are you taking into consideration the opponent's ability to re-roll? The opponent will certainly re-roll a 1, and likely a 2. Assuming your opponent's strategy is to re-roll the 1 and nothing else, if you rolled a 4 your chance of a win would be 5/12, and your chance of a win or a draw would be 7/12. If your opponent re-rolls on 2 as well, your chances get even worse. That being said, I agree that you shouldn't re-roll a 4 although my reasoning is that you are more likely to decrease your roll (3/6) than increase your roll (2/6)
Apr
29
comment Generalized power rule for derivatives
Other than for $f=0$, are there well defined functions $f(x)^{g(x)}$ for which your modified general rule doesn't work?
Apr
29
comment Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$
Why do you specify even constants $c$? Isn't $f^c$ well defined for all integer constants $c$?
Apr
28
comment Generalized power rule for derivatives
@NotNotLogical This is correct. Ill update the question to make that more clear
Apr
28
comment Generalized power rule for derivatives
@Hagen Hmm... I see the problem here. I don't actually care about that case, but I don't really have a good way to tell which case I'm dealing with. I was hoping that there would be some general rule, but I guess Ill have to figure out how to tell what case I'm in, as fgp said.
Apr
28
comment Generalized power rule for derivatives
@Hagen Is that a useful distinction? What is the application of square root if not solving equations? Also, how is that different from the case outlined here, the way it is worded now is "E.g., $y^n=x$ has $n$ solutions in $\mathbb C$" (I'm pointing out the word "solutions" here)
Apr
28
comment Generalized power rule for derivatives
Aren't square roots already non-unique? i.e. $\sqrt{4}=\pm 2$?
Apr
28
comment Generalized power rule for derivatives
Also, I don't actually care about that case
Apr
28
comment Generalized power rule for derivatives
It seems like people are having issues with exponentiation of negative numbers. I'm not really sure how those are usually evaluated, but I can tell you what some of them are. For example: $(-5)^{1/2}=5i$ Other than that, I fear I don't understand what you are asking.
Apr
28
comment Generalized power rule for derivatives
@Hagen Well, I'm talking about derivatives, so if you have defined $f(x)=-1.234$ and $g(x)=4.567$ where $F(x)=f^g$ then $F^\prime(x)=0$
Apr
28
comment Generalized power rule for derivatives
@Hagen I don't understand the question. It looks like you have defined it there. That is one of the cases I don't care about.