Derivatives

A derivative is a rate of change: how fast something is moving at one instant, not averaged across a stretch. This course textbook builds that idea from the ground up, makes it routine to compute, then puts it to work.

Unit 1, Foundations of Derivatives, starts with average speed and narrows it until the average becomes an instant. Six sections cover rates of change, the notion of a limit, what a derivative is, the derivative as a function in its own right, the power rule, and how a derivative reads the shape of a graph.

Unit 2, Derivative Rules and Tests, is the toolkit. Eight sections cover the product, chain and quotient rules, the derivatives of the trig, exponential, logarithmic and inverse trig functions, and the Mean Value Theorem with the first and second derivative tests.

Unit 3, Applications of Derivatives, spends it: implicit differentiation, related rates, optimization, and linear approximation.

It is built to be worked, not just read. Every section opens with a concrete situation and carries the idea all the way through, then hands you practice: 568 numbered exercises across the eighteen teaching sections, the three mixed practice sets, and the algebra appendix, backed by 1,095 hints you can open one at a time when you are stuck and 678 fully worked solutions for when you want to check the whole chain. Each of the three units closes with a recap that checks you against its objectives. Behind the course sit two appendices: twelve algebra refreshers for the skills the course assumes you already have, and a study-tools set with the notation, glossary, methods, formulas, and rules gathered in one place.

Accessibility is built in, not bolted on. All 21,935 mathematical expressions are real MathML carrying a spoken-English label, so VoiceOver reads a formula as a sentence instead of spelling it out symbol by symbol. All 231 instructional diagrams carry a full written description as well as short alternative text, reachable from the figure itself, so a reader who cannot see a diagram can still rebuild it. Text is selectable and resizable, navigation is structural, and the book does not override your reading theme, so White, Sepia, Gray and Night all work as you set them.

The twelve interactive widgets appear here as still images with descriptive captions. The live versions run in the free web edition at derivatives.megan-warren.com.

Written for grades 11 to 12. 2026-2027 edition.