AP Calculus AB / Differentiation: Definition and Fundamental Properties
Lesson
Rules that save you from repeating the limit
Once the definition is honest, power, product, and quotient rules do the daily work.
Learning goals
- Differentiate polynomial and simple rational pieces with the basic rules.
- Explain why a corner can be continuous but not differentiable.
Explanation
The power rule, constant multiple rule, and sum rule come from the limit definition. The product rule is (uv)′ = u′v + uv′. The quotient rule is (u/v)′ = (u′v − uv′) / v². You still need the definition when a problem asks for it, or when a piecewise function’s pieces must match in slope.
If a function is differentiable at a point, it is continuous there. The converse fails: a V-shaped graph is continuous and has no derivative at the point. On a free-response, say “the left- and right-hand derivatives disagree” instead of only “it looks sharp.”
Key terms
- Power rule. The derivative of x^n is n x^{n−1} for the powers used in this course.
- Differentiable. Having a derivative at the point or on the interval in question.
Common mistakes
- Using the product rule but forgetting one of the two terms.
- Canceling across a plus sign in a quotient’s numerator.
Practice
Original Marlow Works items. Check the answer explanation after you try.
If f(x) = 4x³ − 6x + 2, find f′(x).
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Why does continuity of |x| at 0 not give you a derivative there?
Answer. The left-hand slope is −1 and the right-hand slope is 1, so no single f′(0) exists.
Differentiability needs a two-sided limit of difference quotients, not only a connected graph.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source