$x^x$ is my favorite "misbehaving" function. Nearly all exercises, examples and calculations of it involve its relatively tame positive side only. There's also the derivative, $x^x (\ln x +1)$, obviously with log of negatives not having any real value. Everyone seems to discard the left side of the plot for reasons.
Well, I kind of understand. The domain (at least without going into Complex) - as far as I got - seems to be $\Bbb{R}^{+}\cup \Bbb{Z} \cup\text{ {rational fractions with odd denominator} }$. On top of that, it switches the sign depending on whether the numerator of the rational fraction is even or odd. It's either undefined or not a Real number for irrationals and fractions with even denominator.
Then that got me thinking about what its derivative is in that part. Well, duh, obviously there's no derivative because in any arbitrarily short $dx$ it skips an infinite number of times between the positive and negative value. AND it's not continuous! But what if we take only one side of it? Say, $\left| x^x \right|$ ?
Well, obviously still not continuous. But then, we can do the derivative "by definition":
$$ f'(x) = \lim_{h \to 0}{{f(x+h)-f(x)} \over h }$$
providing we limit out $h$ to rational fractions with odd denominator we can get it arbitrarily close to zero, so the limes operator should still work.
WolframAlpha bails out when asked for the plot of the derivative in the negatives, but then if I substitute it with an approximation, I'm getting this peculiar plot:
What would be the analytic function describing its negative part... and how would it relate to the $x^x (\ln x +1)$ from the positive derivative?
Isn't the LHS of your function something like this? $$|(-x)^{-x}|=\left|-\left(\frac{1}{x}\right)^x\right|$$