AP®MathsCollege BoardCalculus ABStudy GuidesUnit 2: Differentiation Definition & Fundamental PropertiesUnit 2 OverviewUnit 2 Summary

Unit 2 Summary (College Board AP® Calculus AB): Study Guide

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 24 April 2026

Differentiation definition & fundamental properties summary

Key definitions

The average rate of change between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is
- $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
The instantaneous rate of change at $x = a$ is
- $\lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
The derivative of the function $f$ is defined by
- $f^{'} (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
A differentiable function is one for which its derivative exists at each point in its domain

Key theorems

Differentiability & continuity
- If a function $f$ is differentiable at $x = a$ , then $f$ is continuous at $x = a$
The product rule
- If $f (x) = g (x) \cdot h (x)$
- Then $f' (x) = g (x) \cdot h' (x) + h (x) \cdot g' (x)$
The quotient rule
- If $f (x) = \frac{g (x)}{h (x)}$
- Then $f' (x) = \frac{h (x) \cdot g' (x) - g (x) \cdot h' (x)}{{[h (x)]}^{2}}$

Key formulas

The derivative of a function multiplied by a constant: If $a$ is a constant
- If $f (x) = a \cdot g (x)$
- Then $f' (x) = a \cdot g' (x)$
The derivative of the sum of two functions:
- If $f (x) = g (x) + h (x)$
- Then $f' (x) = g' (x) + h' (x)$

Derivatives of common functions

$f (x)$	$f' (x)$
$x^{n}$	$n x^{n - 1}$
$e^{x}$	$e^{x}$
$a^{x}$	$a^{x} \cdot \ln a$
$\ln x$	$\frac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$- \sin x$
$\tan x$	$\sec^{2} x$
$\sec x$	$\sec x \tan x$
$\csc x$	$- \csc x \cot x$
$\cot x$	$- \csc^{2} x$

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