Chapter 5: Introductory Differentiation & Integration

Build confidence with the big ideas of calculus: rates at a moment, derivatives, accumulation, integrals, and how they connect.

What You Will Practice

Calculus is built on two major ideas: how fast something changes and how much change accumulates. This lesson introduces differentiation and integration without overwhelming notation.

Differentiation as Rate
Derivative Notation
Simple Derivatives
Integration as Accumulation
Simple Antiderivatives
Differentiation and Integration Connection

Mini Lesson

1. Differentiation as Rate at a Moment

Differentiation describes how fast something is changing at one exact moment.

Example: Velocity is the rate of change of position. Acceleration is the rate of change of velocity.

2. Derivative Notation

Different symbols can represent the derivative.

f′(x) is read as “f prime of x”
dy/dx means “rate of change of y with respect to x”

3. Simple Derivatives

At this level, we preview simple derivatives so calculus notation feels familiar.

d/dx(x) = 1
d/dx(x²) = 2x
d/dx(x³) = 3x²
d/dx(5x) = 5

4. Integration as Accumulation

Integration measures total accumulation. It answers: how much has built up over an interval?

Example: If velocity accumulates over time, the result is distance.

5. Simple Antiderivatives

Integration can reverse differentiation. If the derivative of x² is 2x, then an antiderivative of 2x is x².

∫ x dx = 1/2 x² + C
∫ x² dx = 1/3 x³ + C

6. Connection Between Differentiation and Integration

Differentiation breaks change into rates. Integration rebuilds totals from rates. In formal calculus, this connection is called the Fundamental Theorem of Calculus.

Interactive Intro Calculus Practice

Choose a topic and practice with instant feedback. For derivative answers, type answers like 2x, 3x^2, or 5.

Typing tip: Use simple formats: position, velocity, f prime of x, 2x, x^2+C, or yes.
Big idea: Differentiation tells how fast something changes. Integration tells what all that change adds up to. This is the bridge into Book 3: Calculus I, II, and III.

Mastery Check

Before moving to Book 3, students should be able to recognize these ideas.

Differentiation

I can explain differentiation as instantaneous rate of change.

Notation

I can recognize f′(x) and dy/dx as derivative notation.

Simple Derivatives

I can find basic derivatives such as x² → 2x.

Integration

I can explain integration as accumulation.

Connection

I understand that differentiation and integration undo each other.

Take Book 2 Mastery Check Continue to Book 3 Back to Book 2