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What is a derivative in calculus?

Calculus 1: Master Core Concepts & Applications

Limits

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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

Finding limits from graphs  

Continuity

Discontinuities of Rational Functions (denominator=0)

Finding limits algebraically - direct substitution 

Finding limits algebraically - when direct substitution is not possible

Infinite limits - vertical asymptotes  

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Squeeze theorem

Quadratic Function with Integer Vertex

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits 

Differentiation

Estimating derivatives from a table

Slope and equation of tangent line

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

Derivative of inverse trigonometric functions

Definition of derivative  

Definition of Derivative

Definition of derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

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

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

Implicit differentiation

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

Applications of differentiation

Mean value theorem 

Mean Value Theorem

Mean value theorem

Mean Value Theorem Application

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

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

Optimization

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Curve sketching

Related rates

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

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Calculus 1

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<h2>What is Calculus?</h2>
Providing you with a calculus definition is probably one of the easier things about calculus. A quick google search asking it to define calculus however can quickly become quite a conundrum with the endless results and jargon!
While most intro to calculus courses jump straight into learning the ins and outs of calculus, let&apos;s begin with a brief description of what calculus is &ndash; The study of calculus can be divided into two main branches &ndash; differential calculus and integral calculus. Differential calculus is the focus of Calculus I and involves the study of infinitesimal rates of change. Here you will explore all derivative rules and begin learning applications of calculus that span the disciplines of Physics, Economics, Biology and more. Integral calculus on the other hand, is the study of infinitesimal summation. Integrals are used to identify the sum of individual components on a infinitesimal scale to calculate the whole. Aside from learning integral calculus rules, you will also learn how to apply these rules to calculations of area, volume and displacement. These topics will be discussed in Calculus 2 so don&apos;t panic as we&apos;re only getting started!
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Learning calculus is really not much different as learning algebra or trig. The first step in learning how to do calculus requires you to know how to prepare yourself for calculus:
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<h2>Is calculus hard?</h2>
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<h2>Calc textbooks?</h2>
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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

We have studied the notion of average rate of change thus far, for example, change in position over time (<a href="[[glossary.html|{"keyword":"Velocity"}]]">velocity</a>), average change in velocity over time (acceleration) etc. However, what if we are interested in finding the instantaneous rate of change of something? To answer this, we will first learn about the concept of the definition of derivative in this section, as well as how to apply it.

Composite functions

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

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

Skills

Limits

Derivative Definition

Algebraic Simplification

Binomial Expansion

Conjugates

Rational Expressions

Instantaneous Rate of Change