Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Derivatives
Subtopic: Definition of Derivative

Overview

Calculus I is called Differential Calculus because it's main focus is on differentiating a function resulting in its derivative. This is the process into which we delve today :) In this lesson we introduce derivatives, study the formal definition of derivative, and learn that a derivative is (1) a rate of change, (2) the slope of the tangent line to a curve, and (3) the velocity of a point moving on a curve. Useful stuff in engineering and physical sciences!

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 2.1.1 Understand and be able to apply the formal definition of derivative
• 2.1.2 Understand and be able to apply the alternate definition of derivative
• 2.1.3 Use both definitions of derivative to find the slope of the tangent line to a curve at a point (and find the equation of that tangent line)
• 2.1.4 Recognize graphically points at which a curve is non-differentiable
• 2.1.5 Use both definitions of derivative to find points at which a function is non-differentiable
• 2.1.6 Know that differentiability implies continuity but not the other way around

Terminology

Terms you should be able to define: secant line, tangent line, point of tangency, slope, derivative, rate of change, differentiation, differentiable, non-differentiable, sharp corner, vertical tangent line, cusp

Text Notes

• Several varieties of the formal definition of derivative may be presented including equivalent definitions using different notation (e.g. Newton's vs. Leibnitz's notation) and the "alternate" definition of derivative. Learn to use each of them well. Be mathematically precise with your notation and format.
• Be sure that you understand the graphical relationship between the derivative and the tangent line to a function as well as when a tangent line to a curve does not exist and why.

Mini-Lectures and Examples

Supplemental Resources (optional)

rev. 2020-10-10