# Differentiation: Definition and Fundamental Properties

> AP Calculus BC · Foundations of differential differentiation
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u2-overview/
> Weight: 10-12% of the AP Calculus BC exam

This unit introduces the core definition and rules of differentiation, calculus's foundational tool for measuring rates of change. You will build from first principles to practical rules you will use across the entire course.

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

## Learning objectives

- Define the derivative from first principles as the limit of average rates of change and use standard derivative notation correctly
- Apply core differentiation rules to compute derivatives of basic elementary functions
- Explain the relationship between differentiability and continuity, and identify where functions are not differentiable
- Estimate derivatives at a point from graphs, tables, and numerical data

## Unit at a Glance

This unit follows a logical progression from first principles to practical differentiation. We start by connecting familiar ideas of rate of change to the formal definition of the derivative, then build up the core rules that let you quickly compute derivatives without relying on limits for every calculation.

By the end of the unit, you will be able to differentiate all basic elementary functions, which prepares you for more advanced differentiation techniques and applications of derivatives in later units.

This unit includes the following sub-topics:
- [AP Calculus BC Connecting differentiability and continuity](https://www.owlsprep.com/study/ap-calculus-bc-u2-connecting-differentiability-and-continuity/) — Explore the relationship between differentiability and continuity, including where functions fail to be differentiable.
- [AP Calculus BC Constant, sum, difference, and constant multiple rules](https://www.owlsprep.com/study/ap-calculus-bc-u2-constant-sum-difference-and-constant/) — Learn the most basic differentiation rules for constants, sums, differences, and constant multiples.
- [AP Calculus BC Defining average and instantaneous rates of change at a point](https://www.owlsprep.com/study/ap-calculus-bc-u2-defining-average-and-instantaneous-rates/) — Distinguish between average and instantaneous rates of change and connect them to limits.
- [AP Calculus BC Defining the derivative and using derivative notation](https://www.owlsprep.com/study/ap-calculus-bc-u2-defining-the-derivative-and-using/) — Formalize the definition of the derivative and practice using standard derivative notation.
- [AP Calculus BC Derivatives of cos, sin, e^x, ln(x)](https://www.owlsprep.com/study/ap-calculus-bc-u2-derivatives-of-cos-sin-e/) — Memorize and apply derivative rules for sine, cosine, $e^x$, and $\ln(x)$.
- [AP Calculus BC Derivatives of tan, cot, sec, csc](https://www.owlsprep.com/study/ap-calculus-bc-u2-derivatives-of-tan-cot-sec/) — Learn derivatives of the remaining trigonometric functions: tangent, cotangent, secant, and cosecant.
- [AP Calculus BC Estimating derivatives of a function at a point](https://www.owlsprep.com/study/ap-calculus-bc-u2-estimating-derivatives-of-a-function/) — Estimate the value of a derivative at a point from tables, graphs, and numerical data.
- [AP Calculus BC Power rule](https://www.owlsprep.com/study/ap-calculus-bc-u2-power-rule/) — Apply the power rule, the most widely used rule for differentiating polynomial and power functions.
- [AP Calculus BC Product rule](https://www.owlsprep.com/study/ap-calculus-bc-u2-product-rule/) — Learn how to differentiate products of functions using the product rule.
- [AP Calculus BC Quotient rule](https://www.owlsprep.com/study/ap-calculus-bc-u2-quotient-rule/) — Use the quotient rule to find derivatives of ratios of functions.

## Common pitfalls

- **Wrong:** Assuming all continuous functions are differentiable
  - Why it fails: Continuity does not guarantee differentiability. Functions with corners, cusps, or vertical tangents are continuous but not differentiable.
  - Correct: Always check for sharp points or vertical tangents when confirming differentiability, even if the function is known to be continuous.
- **Wrong:** Misapplying the product or quotient rule formulas
  - Why it fails: Many students incorrectly simplify $(fg)' = f'g'$ or misorder terms in the quotient rule, leading to wrong results.
  - Correct: Memorize the full correct forms: $(fg)' = f'g + fg'$ and $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.
- **Wrong:** Applying the power rule to exponential functions like $e^x$
  - Why it fails: The power rule only applies to functions with constant exponents and variable bases, not the reverse.
  - Correct: Remember that $e^x$ is its own derivative, and only use the power rule for terms like $x^n$.

## Cheatsheet

| Rule/Concept | Formula |
| --- | --- |
| Definition of derivative at $a$ | $f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ |
| Power Rule | $\frac{d}{dx}[x^n] = nx^{n-1}$ |
| Product Rule | $\frac{d}{dx}[f(x)g(x)] = f'g + fg'$ |
| Quotient Rule | $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$ |
| Derivative of $\sin x$ | $\frac{d}{dx} \sin x = \cos x$ |
| Derivative of $\cos x$ | $\frac{d}{dx} \cos x = -\sin x$ |
| Derivative of $e^x$ | $\frac{d}{dx} e^x = e^x$ |
| Derivative of $\ln x$ | $\frac{d}{dx} \ln x = \frac{1}{x} \ (x>0)$ |
| Differentiability implies continuity | If $f$ is differentiable at $a$, it is continuous at $a$ |

## What's next

Begin your study of this unit by learning how average and instantaneous rates of change form the foundation of the derivative. Once you master all the core differentiation rules covered here, you will be ready to move on to more advanced differentiation techniques in the next unit.

- [AP Calculus BC Defining average and instantaneous rates of change at a point](https://www.owlsprep.com/study/ap-calculus-bc-u2-defining-average-and-instantaneous-rates/)
- [AP Calculus BC The chain rule](https://www.owlsprep.com/study/ap-calculus-bc-u3-chain-rule/)
- [Defining the derivative and using derivative notation](https://www.owlsprep.com/study/ap-calculus-bc-u2-defining-the-derivative-and-using/)

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