The engine of the whole course. A derivative is just a slope — but a slope that lives at a single instant, found by watching a secant line pivot into a tangent. Five interactive lessons that turn the limit definition, differentiability, and every fundamental rule into things you can watch happen — the natural next step after limits.
Work them in order for a full walkthrough of the unit, or jump straight to whatever's giving you trouble.
Where the derivative is born. Watch a secant line pivot into a tangent as the interval shrinks to zero — the average rate of change becoming instantaneous. The limit of the difference quotient, the notations for it, and how to read a slope straight off a graph or table.
Every differentiable function is continuous — but not the reverse. See exactly where a derivative fails to exist: the sharp corner, the cusp, the vertical tangent, the jump. The line between a curve that’s locally straight and one that isn’t.
The first shortcuts — and they change everything. Bring down the exponent, drop it by one, and the limit definition you just learned becomes a five-second move. Plus the sum, difference, and constant-multiple rules, horizontal tangents, and the normal line.
The library of derivatives you’ll use forever. Sine and cosine chase each other in a four-step cycle; eˣ is its own derivative; ln x turns into 1/x. Then the four remaining trig functions, built from the quotient rule.
What happens when functions multiply or divide — and why you can’t just differentiate each piece. The product rule’s ‘this-d-that plus that-d-this,’ the quotient rule’s low-d-high pattern, and how to spot which one a problem is asking for.
Put it together the way the real AP exam does — mixed multiple choice first, then full free-response — with a one-page formula sheet to review from.
Thirty multiple-choice questions across all five lessons — the derivative definition, differentiability, the power and basic rules, the key function derivatives, and the product and quotient rules — pulled in fresh batches with worked explanations that name every distractor’s mistake, plus two chained free-response problems in the classic AP Section II mold with typed answers graded against the rubric.
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