The derivative paradox is a problem that happens when you try to figure out the speed of something at a certain moment in time. Imagine you have a car that starts from a standstill, coasts for a few moments, and then stops. The average speed of the car is 0, but at some point during the time when it was coasting, it must have been going a little bit faster than 0. This is the derivative paradox, because if we try to figure out the speed of the car at any one time by taking the derivative (which is a way to measure how much something is changing over time) we get the answer 0 - because the car isn't accelerating, even if it is moving.