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answered simple substitution on equation giving dev a ride…
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comment Derivative of $x^x$ at $x=1$ from first principles
Not sure what you mean by limit of the incremental ratio (I'm really a physicist, so apologies for sloppy nonrigorous math). But just using the definition of derivative: $\displaystyle \lim_{h \rightarrow 0} \frac{(x+h)^{(x+h)} - x^x}{h} = \displaystyle \lim_{h \rightarrow 0} x^x \frac{[(x+h)^h - 1]}{h}$. Since you want this for $x=1$, you have 0/0, so evaluate the limit using L'Hospital's rule at $x = 1$. Apologies if using L'Hospital's rule is assuming too much for whatever you're trying to do?
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answered Derivative of $x^x$ at $x=1$ from first principles
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answered Is there a “most random” state in Rubik's cube?