# KinderCalculus/Speed

We use the term "speed" instead of derivative. Though speed is less general than "rate of change", we prefer "speed" its intuitive appeal; we just need to remind students that it generalizes to other rates. A more mnemonic initialism that serve to remind students of its formal definition from Non-Standard Calculus is "st-ud-r", which stands for STandard of nUDge Ratio -- its formal definition being Standard( ( f(x+h) - f(x) ) / h ). Here, "nudge" is our term for the change in input (input nudge) and change in output (output nudge).

A common pitfall that students assume is that the speed operator distributes. It is helpful to remind students that speed does not "spread" as in the following false formula:

(cf)' = c'f' (c is a constant.) (fg)' = f'g' (f/g)' = f'/g' ( f(g(x)) )' = f'( g'(x) )

By the way, (cf)' = c(f') is called "leak".

## Exercises edit

Aside from the typical computational exercises, students should be able to prove the typical speed identities, especially the Chain & Product Rule. They are difficult, so it's good to do them repeatedly, almost to the point of memorization. This drives home the idea of proofs.