If you take the derivative of, say, xk, you get kxk-1, a function that grows more slowly as x approaches infinity. On the other hand, if you take the derivative of xx, you get xx( ln(x) + 1), a function that grows faster than xx. In fact, if you do this experiment with most standard "nice" functions, you will find that that the derivative makes fastly growing things grow even faster, and slowly growing things grow even slower, with ex as a happy medium whose derivative grows at the same rate as itself. On the other hand, it is not hard to construct "bad" functions that simultaneously grow and wiggle, for which this pattern does not hold. Has anyone looked into this before? I would be interested to know what class of functions behave like this (ie: the derivative is asymptotically smaller than the original function if the original function is asymptotically smaller than ex and the derivative is asymptotically larger if the function is asymptotically larger than ex).