Introduction to the Derivative

AP Calculus · Axiom Academy

INTRO Introduction to the Derivative The slope of a curve at a single point Imagine you're driving a car down a curvy road. Your speedometer tells you how fast you're going at this moment —not your average speed for the whole trip, but your instantaneous speed right now. The same idea applies to mathematical functions. The derivative tells you how fast a function is changing at a specific point. It's the slope of the curve at that exact location. But here's the challenge: a curve doesn't have just one slope like a straight line does. Different parts of the curve are steep or flat. To find the slope at one point , we need a clever idea. Imagine zooming in on the curve with a microscope. The closer you zoom in, the more the curve looks like a straight line. The slope of that line is the derivative. Throughout this unit, you'll learn: The limit definition: The formal, precise mathematical definition Differentiation rules: Shortcuts to find derivatives without using limits every time Applications: How derivatives solve real-world problems about rates of change You've completed the Introduction module! Next: Learn the formal limit definition of the derivative.

This is the written version of the interactive lesson above. See the full AP Calculus course.