## 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.