Some functions are less equal than others

Further evidence, among the piles of assignments I’ve been marking this week, for the suggestion that inequalities can reveal a lot about the depth of students’ understanding:

f(x) \leq g(x) \quad \iff \quad \dfrac{\mathrm{d}f}{\mathrm{d}x} \leq \dfrac{\mathrm{d}g}{\mathrm{d}x}.

Presumably this is just the old fallacy of treating \leq as an equals sign with a fancy haircut, but I recommend it to anyone faced with a tricky analysis proof to complete.

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