Chapter 2.1
Definition of derivative
What You'll Learn
Define the derivative using the limit definition: f'(x) = lim(h0) [f(x+h) - f(x)]/h
Evaluate limits by simplifying rational expressions and canceling common factors
Apply algebraic techniques like binomial expansion, conjugates, and difference of cubes
Identify when derivatives do not exist due to vertical tangent lines or division by zero
Verify derivative results using the power rule as a check
What You'll Practice
1
Finding derivatives of polynomials using the limit definition
2
Using conjugate multiplication to rationalize square root expressions
3
Applying binomial expansion and Pascal's triangle to expand (x+h)³
4
Simplifying complex rational expressions with multiple algebraic steps
5
Determining where derivatives fail to exist and interpreting vertical tangents
Why This Matters
The definition of derivative is the foundation of calculus that connects the concept of instantaneous rate of change to real applications like velocity and acceleration. Mastering this rigorous approach builds your algebra skills and deepens your understanding of why derivative shortcuts like the power rule actually work.
Before You Start — Make Sure You Can:
This Unit Includes
6 Video lessons
Practice exercises
Skills
Limits
Derivative Definition
Algebraic Simplification
Binomial Expansion
Conjugates
Rational Expressions
Instantaneous Rate of Change
