# Differentiation: Definition and Fundamental Properties

> AP Calculus AB · Introduction to derivative definition, core differentiation rules, and the relationship between differentiability and continuity
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u2-overview/
> Weight: 10-12% of AP Calculus AB exam

This unit introduces the definition of the derivative and core rules for computing derivatives, the foundational skill you will use for all AP Calculus AB topics. You will connect derivatives to instantaneous rate of change and learn when derivatives exist.

**Prerequisites:** [Unit 1: Limits and Continuity](https://www.owlsprep.com/study/ap-calculus-ab-u1-overview/)

## Learning objectives

- Define the derivative as the limit of average rates of change and use standard derivative notation correctly
- Explain the relationship between differentiability and continuity, and identify points where a derivative does not exist
- Apply core differentiation rules to compute derivatives of common algebraic and trigonometric functions
- Estimate the derivative of a function at a point from graphs, tables, and numerical data

## 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](https://www.owlsprep.com/study/ap-calculus-ab-u2-connecting-differentiability-and-continuity/) — Explores the relationship between differentiability and continuity, and identifies when a derivative does not exist.
- [AP Calculus AB Constant, sum, difference, and constant multiple rules](https://www.owlsprep.com/study/ap-calculus-ab-u2-constant-sum-difference-and-constant/) — Covers the basic rules for derivatives of constants, sums, differences, and constant multiples.
- [AP Calculus AB Defining average and instantaneous rates of change at a point](https://www.owlsprep.com/study/ap-calculus-ab-u2-defining-average-and-instantaneous-rates/) — Introduces average and instantaneous rates of change as the foundation of the derivative concept.
- [AP Calculus AB Defining the derivative and using derivative notation](https://www.owlsprep.com/study/ap-calculus-ab-u2-defining-the-derivative-and-using/) — Formalizes the definition of the derivative and teaches standard derivative notation conventions.
- [AP Calculus AB Derivatives of cos, sin, e^x, ln(x)](https://www.owlsprep.com/study/ap-calculus-ab-u2-derivatives-of-cos-sin-e/) — Teaches derivative formulas for sine, cosine, $e^x$, and natural logarithm $
(x)$.
- [AP Calculus AB Derivatives of tan, cot, sec, csc](https://www.owlsprep.com/study/ap-calculus-ab-u2-derivatives-of-tan-cot-sec/) — Introduces derivative formulas for the four remaining trigonometric functions.
- [AP Calculus AB Estimating derivatives of a function at a point](https://www.owlsprep.com/study/ap-calculus-ab-u2-estimating-derivatives-of-a-function/) — Shows how to estimate derivatives at a point from graphs, tables, and approximate data.
- [AP Calculus AB Power rule](https://www.owlsprep.com/study/ap-calculus-ab-u2-power-rule/) — Teaches the power rule, the most widely used basic derivative rule for polynomial and power functions.
- [AP Calculus AB Product rule](https://www.owlsprep.com/study/ap-calculus-ab-u2-product-rule/) — Covers the product rule for taking derivatives of products of two differentiable functions.
- [AP Calculus AB Quotient rule](https://www.owlsprep.com/study/ap-calculus-ab-u2-quotient-rule/) — Teaches the quotient rule for computing derivatives of ratios of two differentiable functions.

## Common pitfalls

- **Wrong:** Assuming all continuous functions are differentiable, or that discontinuity does not affect differentiability.
  - Why it fails: Many students mix up the direction of the implication between differentiability and continuity.
  - Correct: Remember that differentiability implies continuity, but continuity does not guarantee differentiability.
- **Wrong:** Reversing the order of terms in the quotient rule numerator.
  - Why it fails: The specific order of terms in the quotient rule is easy to misremember, leading to sign errors.
  - Correct: Memorize the order "low d high minus high d low" or rewrite the quotient as a product to use the product rule.
- **Wrong:** Forgetting the negative sign in the derivative of $\cos x$.
  - Why it fails: The derivative of cosine is the only core trig derivative with a negative sign, so it is often forgotten.
  - Correct: Always double-check the sign when differentiating cosine, cotangent, and cosecant.

## 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 $\implies$ Continuity | Differentiability requires continuity, but continuity does not guarantee differentiability |
| Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$ | Works for all real $n$, including negative and fractional exponents |
| Product Rule: $\frac{d}{dx}[fg] = f'g + fg'$ | Derivative of a product equals the sum of each derivative times the other function |
| Quotient Rule: $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$ | Always keep the order of terms in the numerator to avoid sign errors |
| $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$ | Core derivative rules for the two basic trigonometric functions |
| $\frac{d}{dx}[e^x] = e^x$, $\frac{d}{dx}[\ln x] = \frac{1}{x}$ | Derivatives of the natural exponential and natural logarithm functions |
| $\frac{d}{dx}[\tan x] = \sec^2 x$ | 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.

- [AP Calculus AB Defining average and instantaneous rates of change at a point](https://www.owlsprep.com/study/ap-calculus-ab-u2-defining-average-and-instantaneous-rates/)
- [Unit 3: Differentiation: Composite, Implicit, and Inverse Functions Overview](https://www.owlsprep.com/study/ap-calculus-ab-u3-overview/)
- [Defining the derivative and using derivative notation](https://www.owlsprep.com/study/ap-calculus-ab-u2-defining-the-derivative-and-using/)

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