Jim-Bob Shmoopla is speeding along in his Indycar. He's wanted to win this particular race since his older sister, Jillly-Lou, won it five years prior. Not only does he want to win, he wants to beat her highest speed of 240 miles per hour for a single lap during the qualifier. He also wants to reach an instantaneous speed of 250 mph, way better than Jilly ever did. He knows it's impossible for himself to maintain a speed of 250 mph for more than a moment, but he wants to have bragging rights nonetheless. Not to mention the twist soft-serve cone she promised him if he could achieve both. Lucky for Jimmy-B, he can use his speedometer to figure this out. Even luckier, his speedometer knows the basics of calculus.
Speed is an example of a derivative. In fact, it's the slope of the distance vs. time plot. Without calculus, Jim-Bob could possibly tell if he beat his sister's single lap time. He knows the distance of a lap, and a stopwatch is easy to come by. However, it would be hard (or impossible) to tell if he ever reached an instantaneous speed of 250 mph without calculus.
As Jim-Bob heads for the home stretch, he and his crew are thrilled to see the pin brush the 250 mph mark. Victory is in his reach; he can taste the creamy goodness already. He crosses the finish line and meets with his pit crew. They give him the bad news that his average speed over the lap was 235 mph. Better luck next time, Jim-Bob. Jilly-Lou Shmoopla's family record lives on.
In the most general terms, the derivative of a function is the slope of that function. Since most functions aren't straight lines, the definition of "slope" takes a little more work than it did before.
There are a couple of different ways to approach derivatives. We can go about it in a more math-y way, or we can go about it in a more physics-y way.
Once we know what derivatives are, we can do some interesting stuff including relating derivatives to the functions from which they came.