Unit 2 · Core unit

The Derivative and Differentiation Rules

The derivative is defined as a limit and then made practical by rules. Building it from the difference quotient once is what makes the rules feel like shortcuts rather than magic.

What a strong answer looks like

A strong answer names the rule being applied and, for the chain rule, states what the inner function is before differentiating.

Practice this unit Course overview

Chapter introduction

Understand the idea before memorizing the rule

The derivative is defined as a limit and then made practical by rules. Building it from the difference quotient once is what makes the rules feel like shortcuts rather than magic.

In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

The Derivative as a Limit and as a Rate

Concept explanation

The difference quotient measures the slope of a secant; its limit is the tangent slope, which is also the instantaneous rate of change with units attached.

How to apply it

Compute one derivative from the definition each time a new family appears, then use the rules.

Common mistake

Reporting a derivative as a number with no interpretation, or confusing f(a) with f prime of a.

Guided study move

Compute the derivative of a quadratic from the definition, then confirm it with the power rule.

Lesson 2

Choosing Among the Rules

Concept explanation

Product, quotient, and chain rules each answer a different structural question: is the expression a product, a quotient, or a composition?

How to apply it

Describe the structure of the expression out loud before differentiating, then apply the matching rule.

Common mistake

Differentiating factors separately and multiplying, or forgetting the derivative of the inner function.

Guided study move

Sort ten expressions by which rule they need before differentiating any of them.

Lesson 3

Differentiability and Implicit Differentiation

Concept explanation

Differentiability implies continuity but not the reverse; corners and vertical tangents break it. Implicit differentiation treats y as a function of x throughout.

How to apply it

Differentiate both sides with respect to x, attach dy/dx wherever y is differentiated, then solve for it.

Common mistake

Assuming a continuous graph must be differentiable, or forgetting dy/dx on the y terms.

Guided study move

Find the tangent line to a circle at a point using implicit differentiation.

Formula and visual reference

Connect the symbols to the picture

Chapter synthesis

Ideas to connect

Naming the structure before the rule
Keeping the chain rule visible in every composition
Attaching units to a derivative in context

Modeling lab

Transfer the chapter to a new setting

A quantity is given by a composition of two contextual functions. Differentiate it, then state what the result measures and in what units.

Mastery checklist

• Derive a derivative from the difference quotient.
• Differentiate a product, a quotient, and a composition.
• Explain why the absolute value function is not differentiable at zero.
• Find dy/dx implicitly and evaluate it at a point.

Check yourself

• Why does differentiability imply continuity but not the reverse?
• What is the inner function, and why must its derivative appear?

Modeling drill

A quantity is given by a composition of two contextual functions. Differentiate it, then state what the result measures and in what units.

Difference quotientTangent lineInstantaneous rate of changeProduct ruleQuotient ruleChain ruleImplicit differentiationDifferentiability

Unit 1: Limits and Continuity

Unit 3: Applications of Differentiation