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

Introduction

The derivative begins as a limit of average rates and becomes a function that records local slope at every differentiable input.

Elementary derivative rules preserve coefficients, domains, angle conventions, and the separate contributions of products and quotients.

A correct derivative includes its mathematical expression, valid inputs, and a rate interpretation when the variables carry units.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; series, parametric, polar, and vector reasoning remain part of the BC review.

2.1

Defining Average and Instantaneous Rates of Change at a Point

How does an average rate become a rate at one point?

Short answer

Take the limit of secant slopes as the second input approaches the fixed input; the limiting slope is the derivative there.

Know firstAverage rate · Limits · Units

Keep the rate meaning while the interval shrinks

Average change over [a,a+h] is [f(a+h)−f(a)]/h. It measures output change per input change and therefore carries rate units. The derivative asks whether these average rates stabilize as h approaches zero from both sides.

For f=x²+2x at a=1, the quotient simplifies to 4+h. Its limit is 4. That number is not f(1)=3; it describes how quickly f is changing near x=1.

Formula

Formula and conditions

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

The derivative value is the limiting average rate.

Use when: The finite two-sided limit exists; use an inward derivative only when explicitly working at a domain endpoint.
Do not assume: Function value and derivative value answer different questions.

Secant slopes approach a local rate

For f=x²+2x at x=1, the secant slope is 4+h. Moving the second input toward 1 makes that average rate approach the tangent slope 4.

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

Secant slopes approach a local rate: values

Case	Result
$h=1$	secant slope 5
$h=0.1$	4.1
$h=-0.1$	3.9
limit	4

Differentiate with the structure visible

Write the secant quotient.
Simplify before setting h near zero.
Take the two-sided limit.
Attach output-per-input units.

Keep these three ideas

Secant gives average rate.
Limit gives tangent rate.
Derivative keeps 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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