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Calculus I · Unit 2A · hub

Unit 2A: Derivative Foundations and Differentiation Techniques

Study Unit 2A: Derivative Foundations and Differentiation Techniques through a complete sequential course with explanations, visuals, checks, reviews, practice exams, and answer keys.

Core textbook pathChoose the next lesson from the ordered unit map.

Core textbook

The complete Unit 2A path

Start with meaning, build the differentiation toolkit in sequence, and use reviews to make rule selection independent of page labels.

What this unit teaches

Turn local change into a dependable derivative toolkit.

Connect limits, tangent slopes, formulas, graphs, tables, and units; then differentiate powers, products, quotients, special functions, compositions, implicit equations, inverses, and variable exponents.

Prerequisites

Algebra, functions, and Unit 1 limits.

You should be comfortable with factoring, exponents, function notation, slopes, and finite limits. Use the diagnostic when you are unsure.

Section

Orientation and prerequisites

Treat the derivative as a local response rate before treating it as a formula.

  1. 01Unit 2A: Derivative Foundations and Differentiation Techniqueshub
  2. 02What Does the Word Derivative Mean in Calculus?lesson
Section

Derivative meaning and foundations

Watch a secant slope stabilize into a tangent slope and then generalize from one point to a derivative function.

  1. 03Why Derivatives Matterlesson
  2. 04The Derivative at a Pointlesson
  3. 05Difference Quotient Algebra Without Skipped Stepslesson
  4. 06Tangent and Normal Lineslesson
  5. 07The Derivative as a Functionlesson
  6. 08Derivative Notation and Unitslesson
  7. 09Estimating Derivatives from Graphs and Tableslesson
  8. 10Differentiability and Continuitylesson
  9. 11Derivative Foundations Reviewreview
Section

Core differentiation rules

Every rule is a compressed limit calculation; choose the structure before doing algebra.

  1. 12The Constant and Power Ruleslesson
  2. 13Derivative Rules Are Shortcuts, Not New Definitionslesson
  3. 14Negative and Fractional Powerslesson
  4. 15Sums and Constant Multipleslesson
  5. 16The Product Rulelesson
  6. 17The Quotient Rulelesson
  7. 18How to Choose a Differentiation Rulelesson
  8. 19Core Differentiation Rules Reviewreview
Section

Trigonometric, exponential, and logarithmic functions

Connect each derivative formula to the function's characteristic growth, periodicity, or inverse relationship.

  1. 20Derivatives of Sine and Cosinelesson
  2. 21Why Special-Function Derivatives Are Not Randomlesson
  3. 22Derivatives of the Other Trigonometric Functionslesson
  4. 23The Derivative of the Natural Exponentiallesson
  5. 24Derivatives of General Exponential Functionslesson
  6. 25The Derivative of the Natural Logarithmlesson
  7. 26Derivatives of General Logarithmslesson
  8. 27Trigonometric, Exponential, and Logarithmic Reviewreview
Section

The chain rule and compositions

Read nested functions from the outside inward, but multiply local response factors through every layer.

  1. 28Composition and the Need for the Chain Rulelesson
  2. 29A Visual and Verbal Map of the Chain Rulelesson
  3. 30The Basic Chain Rulelesson
  4. 31Multiple Chain-Rule Layerslesson
  5. 32Chain Rule with Trigonometric Functionslesson
  6. 33Chain Rule with Exponentials and Logarithmslesson
  7. 34Combining Product, Quotient, and Chain Ruleslesson
  8. 35Chain Rule Reviewreview
Section

Implicit, inverse, and logarithmic differentiation

Track which variable depends on which and use reciprocal or logarithmic structure only where its conditions hold.

  1. 36Implicit Differentiationlesson
  2. 37What “Implicit” Means and Why It Matterslesson
  3. 38Tangent Lines to Implicit Curveslesson
  4. 39Second Derivatives from Implicit Equationslesson
  5. 40Derivative of an Inverse Functionlesson
  6. 41Inverse Trigonometric Derivativeslesson
  7. 42Logarithmic Differentiationlesson
  8. 43Variable Bases and Exponentslesson
  9. 44Implicit, Inverse, and Logarithmic Differentiation Reviewreview
Section

Higher derivatives and complete strategy

Treat repeated derivatives as repeated questions about change, not as superscripts to manipulate mechanically.

  1. 45Higher Derivativeslesson
  2. 46How to Compute Higher Derivatives Reliablylesson
  3. 47A Complete Derivative Computation Strategylesson
Section

Review, practice, exams, and reference

Mixed practice tests recognition: the page title no longer tells you which rule to use.

  1. 48Unit 2A Review: Derivative Meaning and Computationreview
  2. 49Common Derivative Computation Errors and How to Repair Themreview

Practice around the path

Reviews, quizzes, diagnostics, and exams

Use these after a section or whenever a worked example reveals a specific gap. The answer keys are separated so an honest first attempt stays easy.

diagnosticDerivative Prerequisite DiagnosticCheck the prerequisite skills needed for Derivative Foundations and Techniques and follow targeted review links before beginning the unit.referenceDerivative Notation GuideUse this derivative foundations and techniques reference for formulas, notation, domains, units, and problem-solving strategy.reviewDerivative Foundations ReviewReview derivative foundations review with mixed practice and links back to the exact lessons behind each skill.quizDerivative Foundations Concept QuizTest conceptual understanding of derivative foundations concept quiz with a short self-check and model responses.reviewCore Differentiation Rules ReviewReview core differentiation rules review with mixed practice and links back to the exact lessons behind each skill.quizCore Rules Concept QuizTest conceptual understanding of core rules concept quiz with a short self-check and model responses.reviewTrigonometric, Exponential, and Logarithmic ReviewReview trigonometric, exponential, and logarithmic review with mixed practice and links back to the exact lessons behind each skill.quizSpecial Functions Concept QuizTest conceptual understanding of special functions concept quiz with a short self-check and model responses.reviewChain Rule ReviewReview chain rule review with mixed practice and links back to the exact lessons behind each skill.quizChain Rule Concept QuizTest conceptual understanding of chain rule concept quiz with a short self-check and model responses.reviewImplicit, Inverse, and Logarithmic Differentiation ReviewReview implicit, inverse, and logarithmic differentiation review with mixed practice and links back to the exact lessons behind each skill.quizImplicit and Inverse Concept QuizTest conceptual understanding of implicit and inverse concept quiz with a short self-check and model responses.reviewUnit 2A Review: Derivative Meaning and ComputationReview unit 2a review: derivative meaning and computation with mixed practice and links back to the exact lessons behind each skill.reviewCommon Derivative Computation Errors and How to Repair ThemReview common derivative computation errors and how to repair them with mixed practice and links back to the exact lessons behind each skill.practiceUnit 2A Cumulative Derivative PracticePractice unit 2a cumulative derivative practice with mixed problems, staged guidance, and source-linked solutions.examUnit 2A Practice Exam AComplete Unit 2A Practice Exam A as a cumulative Calculus I assessment, then use the separately published answer key for review.examUnit 2A Practice Exam BComplete Unit 2A Practice Exam B as a cumulative Calculus I assessment, then use the separately published answer key for review.referenceUnit 2A Formula and Strategy ReferenceUse this derivative foundations and techniques reference for formulas, notation, domains, units, and problem-solving strategy.

Check your work

Published exam answer keys

Every exam has a separately routed, numbered key. Finish the exam first, then compare one item at a time.

Go deeper

Focused integral explorations

These articles zoom in on one derivative pattern, proof idea, or decision. They are enrichment around the textbook path, not a replacement for it.

Advanced explorationAdvanced Notes: What Differentiability Is Really SayingExplore advanced notes: what differentiability is really saying in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationDifferentiability, Little-o Notation, and the Best Local Linear ModelExplore differentiability, little-o notation, and the best local linear model in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationDarboux's Theorem: Derivatives Cannot JumpExplore darboux's theorem: derivatives cannot jump in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationThe Chain Rule as Composition of Local Linear MapsExplore the chain rule as composition of local linear maps in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationA Preview of the Implicit Function TheoremExplore a preview of the implicit function theorem in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Focused articleWhy x to the x needs logarithmic differentiationA concise in-depth exploration connected to the main unit.Focused articleThe Chain Rule as a structure-reading skillA concise in-depth exploration connected to the main unit.Focused articleProduct Rule or Quotient Rule?A concise in-depth exploration connected to the main unit.Focused articleInverse-trigonometric derivative patternsA concise in-depth exploration connected to the main unit.

Next in the textbook

Unit 2B: Applications of Derivatives

Put derivative calculations to work in motion, approximation, related rates, curve analysis, optimization, indeterminate limits, and applied modeling.

Continue to Unit 2B →

Unit 2A Hub: Derivative Foundations and Differentiation Techniques

A derivative is a local response rate. It tells how fast an output is changing, how steep a graph is, or how sensitive a model is at a particular input. This unit develops that idea from limits and then builds a complete, dependable differentiation toolkit.

In ordinary language

In ordinary language

If a function answers "how much?", its derivative answers "how fast is that amount changing right here?" The function and derivative are different functions with different jobs.

Main sequence

• Prerequisite diagnostic and notation guide. • The derivative as a shrinking-interval limit. • Derivatives at points, tangent lines, and derivative functions. • Meaning, units, graph and table interpretation, and differentiability. • Power, sum, product, and quotient rules. • Trigonometric, exponential, and logarithmic derivatives. • Chain rule and multi-rule expressions. • Implicit, inverse, and logarithmic differentiation. • Higher derivatives and a complete computation strategy. • Reviews, concept quizzes, cumulative practice, and two exams.

What is deliberately postponed

Motion analysis, related rates, linear approximation, graph-shape theorems, optimization, and L'Hopital's Rule belong to Unit 2B. Small applications still appear here because formulas are easier to remember when they have a reason to exist, but the main task of 2A is calculation and meaning.

Source note

The sequence and activity-first pedagogy were informed by Active Calculus, Single Variable, 2nd Edition. Intuitive and historical perspectives were informed by Silvanus P. Thompson's Calculus Made Easy. Traditional exercise patterns were checked against the public-domain Granville--Smith and Greenhill texts. BetterGrades prose, examples, exercises, route structure, and graph specifications are independently written unless explicitly attributed.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.