Unit Overview

Differentiation: Definition and Fundamental Properties

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

1. 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

Explore the relationship between differentiability and continuity, including where functions fail to be differentiable.

★★⏱ 4 min

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

Learn the most basic differentiation rules for constants, sums, differences, and constant multiples.

★⏱ 3 min

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

Distinguish between average and instantaneous rates of change and connect them to limits.

★★⏱ 4 min

AP Calculus BC Defining the derivative and using derivative notation

Formalize the definition of the derivative and practice using standard derivative notation.

★★⏱ 4 min

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

Memorize and apply derivative rules for sine, cosine, $e^{x}$ , and $l n (x)$ .

★★⏱ 4 min

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

Learn derivatives of the remaining trigonometric functions: tangent, cotangent, secant, and cosecant.

★★★⏱ 4 min

AP Calculus BC Estimating derivatives of a function at a point

Estimate the value of a derivative at a point from tables, graphs, and numerical data.

★★⏱ 3 min

AP Calculus BC Power rule

Apply the power rule, the most widely used rule for differentiating polynomial and power functions.

★⏱ 3 min

AP Calculus BC Product rule

Learn how to differentiate products of functions using the product rule.

★★⏱ 4 min

AP Calculus BC Quotient rule

Use the quotient rule to find derivatives of ratios of functions.

★★★⏱ 4 min

2. Common Pitfalls

Wrong move:

Assuming all continuous functions are differentiable

Why:

Continuity does not guarantee differentiability. Functions with corners, cusps, or vertical tangents are continuous but not differentiable.

Correct move:

Always check for sharp points or vertical tangents when confirming differentiability, even if the function is known to be continuous.

Wrong move:

Misapplying the product or quotient rule formulas

Why:

Many students incorrectly simplify $(f g)^{'} = f^{'} g^{'}$ or misorder terms in the quotient rule, leading to wrong results.

Correct move:

Memorize the full correct forms: $(f g)^{'} = f^{'} g + f g^{'}$ and $(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ .

Wrong move:

Applying the power rule to exponential functions like $e^{x}$

Why:

The power rule only applies to functions with constant exponents and variable bases, not the reverse.

Correct move:

Remember that $e^{x}$ is its own derivative, and only use the power rule for terms like $x^{n}$ .

3. Quick Reference Cheatsheet

Rule/Concept	Formula
Definition of derivative at $a$	$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$
Power Rule	$\frac{d}{d x} [x^{n}] = n x^{n - 1}$
Product Rule	$\frac{d}{d x} [f (x) g (x)] = f^{'} g + f g^{'}$
Quotient Rule	$\frac{d}{d x} [\frac{f}{g}] = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$
Derivative of $sin x$	$\frac{d}{d x} sin x = cos x$
Derivative of $cos x$	$\frac{d}{d x} cos x = - sin x$
Derivative of $e^{x}$	$\frac{d}{d x} e^{x} = e^{x}$
Derivative of $ln x$	$\frac{d}{d x} ln x = \frac{1}{x} (x > 0)$
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.