The Derivative Summary

Calculus 1 · Axiom Academy

Everything from Unit 2 in one place: the derivative as instantaneous rate of change, born from a limit and read as the slope of a tangent line. The derivative measures an instantaneous rate of change — how fast f is changing at a single point, not over an interval. It is defined as a limit of the difference quotient : . Geometrically, f'(a) is the slope of the tangent line — the limit of secant slopes as the second point slides into the first. The same number, f'(a) , answers physical questions: if s(t) is position, s'(t) is velocity. Where a graph has a corner, cusp, vertical tangent, or break, the derivative does not exist . Core Concept Average vs. Instantaneous Rate The average rate of change over an interval is the slope of the secant line through two points. Shrinking the interval ( ) turns that average into the instantaneous rate at one point. Average: needs two points, a whole interval of width h . Instantaneous: the limit as the interval collapses to a single point. Core Concept The Limit Definition This is the heart of the unit: the derivative is a limit. The difference quotient is the slope of a secant; taking gives the slope of the tangent — provided that limit exists. Watch out for: the limit must agree from both sides, or f' does not exist there. Pin one point and let a second point slide toward it. Each pair gives a secant line ; as the gap h vanishes, the secants pivot into the single tangent line whose slope is the derivative.

This is the written version of the interactive lesson above. See the full Calculus 1 course.