What's a derivative? (Where does it come from and what is it used for?)

(aka my way of seeing if they're paying attention in class or just using a formula)

This girl's answer:

A derivative is a simplified form of an equation that can be derived from the original and can be used to find the answers for different equations.

Um, no.

She had a lot more issues than this... every single one of her difference quotient expressions had x -> 0.

This girl also cried after my first semester exam. Apparently she didn't learn a lot from that experience.

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