AP Calculus AB Unit 2: Differentiation: Definition and Fundamental Properties Notes

Introduction

The derivative turns the limit of average rates into a function that records local change. Graphs, tables, equations, and units each reveal a different part of that definition.

The basic derivative rules are compressed consequences of the limit definition. Linearity handles sums, while product and quotient rules account for multiple changing factors.

Differentiability is stronger than continuity. Corners, cusps, vertical tangents, endpoints, and discontinuities explain exactly where a derivative function loses inputs.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; the reader marks calculator decisions separately from mathematical justification.

2.1

Defining Average and Instantaneous Rates of Change at a Point

How are average and instantaneous rates related but not interchangeable?

Short answer

Average rate is a secant slope across an interval; instantaneous rate is the limiting secant slope as the interval collapses to one input.

Know firstSlope · Function notation · Limits

For inputs a and b, the average rate of change is $[f(b)-f(a)]/(b-a)$ . It reports one net output change per input unit across the whole interval. Different behavior inside the interval can produce the same average.

To measure local behavior at a, write b=a+h and let h approach zero. The second point remains distinct while the secant line rotates toward a tangent line. If its slope has one finite limit, that limit is the instantaneous rate. Average and instantaneous rates can coincide, but one does not generally determine the other.

Formula

Formula and conditions

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}

The derivative is a limiting average rate.

Use when: The quotient is evaluated for h≠0 and has one finite two-sided limit.
Do not assume: Substituting h=0 before simplifying creates 0/0.

Secant slope approaching the tangent slope

For f(x)=x² at x=1, the secant slope is 2+h and approaches the tangent slope 2 as h approaches zero.

The interactive version of this figure is part of the full study guide.

Secant slope approaching the tangent slope: values

Case	Result
$h=2$	secant slope $4$
$h=1$	secant slope $3$
$h=0.1$	secant slope $2.1$

Use the idea

Use the interval endpoints for an average rate.
Use a difference quotient and a limit for an instantaneous rate.
Interpret the sign and units after computing the slope.

Keep these three ideas

Secant means interval.
Tangent means local limit.
Both slopes have rate units.

Keep reading Unit 2: Differentiation: Definition and Fundamental Properties

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2.1 Defining Average and Instantaneous Rates of Change at a PointRead

2.2 Defining the Derivative of a Function and Using Derivative Notation

2.3 Estimating Derivatives of a Function at a Point

2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

2.5 Applying the Power Rule

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

2.7 Derivatives of cos x, sin x, e^x, and ln x

2.8 The Product Rule

2.9 The Quotient Rule

2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

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