AP Calculus AB / Differentiation: Definition and Fundamental Properties
Lesson
From secant to tangent
Shrink the interval in an average rate and you are looking at a derivative.
Learning goals
- Write and interpret a difference quotient.
- Connect f′(a) to a tangent slope and to a rate in context.
Explanation
The average rate of change of f on [a, a + h] is [f(a + h) − f(a)] / h. That is the slope of a secant. The derivative f′(a) is the limit of those slopes as h approaches 0, when the limit exists. Instantaneous is a limit of averages, not a different species of number.
If s(t) is position in meters and t is seconds, s′(t) is meters per second. A positive derivative means the output is increasing. Zero can be a pause or a peak; the sign chart later tells you which. A sharp corner, as on |x| at 0, has no single tangent slope.
Key terms
- Difference quotient. The average rate [f(a + h) − f(a)] / h on an interval of width h.
- Derivative at a point. The limit of difference quotients as the interval width goes to zero, if that limit exists.
Common mistakes
- Forgetting to take the limit and calling a secant slope “the derivative.”
- Dropping units when the function is a real-world quantity.
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
A car’s odometer goes from 40 km to 70 km between t = 1 h and t = 2 h. Is 30 km/h a derivative?
Answer. It is an average rate, not an instantaneous derivative at a single time.
A derivative would be the limit of such averages on shrinking intervals around one t.
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