One easy way to show that more generally is that if f and g are differentiable, f(x) dominates g(x) iff f'(x) dominates g'(x), by L'Hopital's rule. Details left as an exercise for the reader.
Ah, yes, right. Smoothness alone doesn't do it. Nonetheless, I still maintain that the property holds for all functions that we actually care about when doing analysis of algorithms. In particular, it certainly holds for sums and products of n! e^n, n log n, n, log n, and 1, which covers probably 99.9% of everything I've ever seen inside an O().
Anyway you can show that on the spot if you think about what the derivative of that monomial looks like