Unit Overview

Differentiation: Definition and Fundamental Properties

AP Calculus AB· 5 min read 📊 10-12% of AP Calculus AB exam

1. Unit at a Glance

This unit builds on the limits you learned in Unit 1 to formalize the concept of instantaneous rate of change, which is the core of differentiation. We start with the formal definition of the derivative, then build up basic rules that let you skip tedious limit calculations for most common functions, turning derivative computation into a straightforward algebraic skill.

The unit progresses from conceptual understanding to rule application, so you will first master what a derivative is before learning the shortcuts you will use for the rest of the course. All subsequent units in AP Calculus AB depend on the skills you build in this unit.

This unit covers the following key sub-topics:

AP Calculus AB Connecting differentiability and continuity

Explores the relationship between differentiability and continuity, and identifies when a derivative does not exist.

★★⏱ 8 min

AP Calculus AB Constant, sum, difference, and constant multiple rules

Covers the basic rules for derivatives of constants, sums, differences, and constant multiples.

★⏱ 6 min

AP Calculus AB Defining average and instantaneous rates of change at a point

Introduces average and instantaneous rates of change as the foundation of the derivative concept.

★★⏱ 7 min

AP Calculus AB Defining the derivative and using derivative notation

Formalizes the definition of the derivative and teaches standard derivative notation conventions.

★★⏱ 8 min

AP Calculus AB Derivatives of cos, sin, e^x, ln(x)

Teaches derivative formulas for sine, cosine, $e^{x}$ , and natural logarithm $ (x)$.

★★⏱ 7 min

AP Calculus AB Derivatives of tan, cot, sec, csc

Introduces derivative formulas for the four remaining trigonometric functions.

★★★⏱ 6 min

AP Calculus AB Estimating derivatives of a function at a point

Shows how to estimate derivatives at a point from graphs, tables, and approximate data.

★★★⏱ 7 min

AP Calculus AB Power rule

Teaches the power rule, the most widely used basic derivative rule for polynomial and power functions.

★⏱ 6 min

AP Calculus AB Product rule

Covers the product rule for taking derivatives of products of two differentiable functions.

★★⏱ 7 min

AP Calculus AB Quotient rule

Teaches the quotient rule for computing derivatives of ratios of two differentiable functions.

★★★⏱ 7 min

2. Common Pitfalls

Wrong move:

Assuming all continuous functions are differentiable, or that discontinuity does not affect differentiability.

Why:

Many students mix up the direction of the implication between differentiability and continuity.

Correct move:

Remember that differentiability implies continuity, but continuity does not guarantee differentiability.

Wrong move:

Reversing the order of terms in the quotient rule numerator.

Why:

The specific order of terms in the quotient rule is easy to misremember, leading to sign errors.

Correct move:

Memorize the order "low d high minus high d low" or rewrite the quotient as a product to use the product rule.

Wrong move:

Forgetting the negative sign in the derivative of $cos x$ .

Why:

The derivative of cosine is the only core trig derivative with a negative sign, so it is often forgotten.

Correct move:

Always double-check the sign when differentiating cosine, cotangent, and cosecant.

3. Quick Reference Cheatsheet

Concept / Formula	Key Summary
Derivative at $a$ : $f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$	Formal limit definition of the instantaneous rate of change at $x = a$
Differentiability $⟹$ Continuity	Differentiability requires continuity, but continuity does not guarantee differentiability
Power Rule: $\frac{d}{d x} [x^{n}] = n x^{n - 1}$	Works for all real $n$ , including negative and fractional exponents
Product Rule: $\frac{d}{d x} [f g] = f^{'} g + f g^{'}$	Derivative of a product equals the sum of each derivative times the other function
Quotient Rule: $\frac{d}{d x} [\frac{f}{g}] = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$	Always keep the order of terms in the numerator to avoid sign errors
$\frac{d}{d x} [sin x] = cos x$ , $\frac{d}{d x} [cos x] = - sin x$	Core derivative rules for the two basic trigonometric functions
$\frac{d}{d x} [e^{x}] = e^{x}$ , $\frac{d}{d x} [ln x] = \frac{1}{x}$	Derivatives of the natural exponential and natural logarithm functions
$\frac{d}{d x} [tan x] = sec^{2} x$	Commonly tested derivative rule for the tangent function

What's Next

To begin this unit, start with the conceptual foundation of derivatives as rates of change. This unit builds sequentially from definition to rule application, so completing sub-topics in order will help you connect concepts to the computational skills you need for the AP exam. Once you finish all sub-topics in this unit, you will be ready to move on to more advanced differentiation topics in the next unit.