About Us

Why Choose StudyPug

How it Works

Pricing

Testimonials

Contact Us

Blog

Basic Math

Algebra

Geometry

Trigonometry

Calculus

Linear Algebra

Differential Equations

Statistics

Chemistry

Organic Chemistry

Physics

Microeconomics

For Students

For Parents

For Home Schoolers

For Teachers

Australia

Canada

Ireland

New Zealand

Singapore

United Kingdom

United States

México

Latinoamérica

Sitemap

Terms of Service

Privacy Policy

Accessibility

Definition of Derivative: The Limit Definition Explained

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

auto-gen

Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Parent Assignments

Challenge Zone

Foundation Review

math

Finding the derivative of a cubic polynomial using the limit definition

Differentiating a square root function using the limit definition

Differentiating a rational function using the limit definition

Differentiating a cube root function using the definition of derivative

Identifying where the vertical tangent line occurs at x = 0

Showing the derivative does not exist at x = 0

The definition of derivative describes the instantaneous rate of change of a function, built from the limit of a difference quotient as the interval shrinks to zero. This page walks through the formula, why it works, and how to apply it to real functions with worked examples.

Definition of derivative  

Composite functions

MY PROGRESS

Pug Score

Get Started

Topic Progress

Pug Score

Overview

Definition of Derivative: The Limit Definition Formula

What You'll Learn

The definition of derivative is built from a limit, not just a formula to memorize, so understanding the limit process is essential.

The derivative of f(x) at a point equals the limit of the difference quotient as h approaches zero.

This limit represents the slope of the tangent line to the curve at that exact point.

Working through the algebra of the difference quotient before taking the limit is the key skill in every example.

Once the limit definition is understood, shortcut rules make finding derivatives much faster for everyday problems.

What You'll Practice

Finding derivatives of polynomials using the limit definition

Using conjugate multiplication to rationalize square root expressions

Applying binomial expansion and Pascal's triangle to expand (x+h)³

Simplifying complex rational expressions with multiple algebraic steps

Determining where derivatives fail to exist and interpreting vertical tangents

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

6 Video lessons

Practice exercises

Learning resources

Skills

Limits

Derivative Definition

Algebraic Simplification

Binomial Expansion

Conjugates

Rational Expressions

Instantaneous Rate of Change