Limits

Introduction to Limits

Properties of Limits

Formal Definition of Limits Part 1

Formal Definition of Limits Part 2

Ex: Limit Definition - Find Delta Values, Given Epsilon For a Limit

Ex 1: Limit Definition - Determine Delta for an Arbitrary Epsilon (Linear)

Ex 2: Limit Definition - Determine Delta for an Arbitrary Epsilon (Quadratic)

Determining Limits

Ex 1: Determine a Limit Numerically

Ex 2: Determine a Limit Numerically

Ex 3: Determine a Limit Numerically

Examples: Determining Basic Limits Graphically

Ex 1: Determine Limits from a Given Graph

Ex 2: Determine Limits from a Given Graph

Determine Special Limits Using a Table and a Graph

Determine Limits and One-sided Limits by Analyzing a Table of Values and a Graph

Ex 1: Determine Limits from a Graph Using Function Notation

Ex 2: Determine Limits from a Graph Using Function Notation (Challenging)

Ex: Determining Basic Limits Using Direct Substitution

Ex: Determining Limits Involving an Absolute Value Function Graphically and Algebraically

Ex 1: Determining Limits and One-Sided Limits Graphically

Ex 2: Determining Limits and One-Sided Limits Graphically

Determine Limits and One-Sided Limits from a Graph

Ex 1: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 2: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 3: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 4: One-Sided Limits and Vertical Asymptotes (Tangent Function)

Ex 5: One-Sided Limits and Vertical Asymptotes (Cosecant Function)

Ex 1: Determine a Limit Analytically

Ex: Limits Involving the Greatest Integer Function

Ex: Limits of the Floor Function (Greatest Integer Function)

Ex: Determining Limits of Rational Functions by Factoring

Ex 2: Determine a Limit of a Piece-Wise Defined Function Analytically

Ex 3: Determine a Limit Analytically by Factoring

Ex 4: Determine Limits of a Rational Function Analytically

Ex 1: Determine a Limit of a Rational Function by Expanding or Factoring

Ex 2: Determine a Limit of a Rational Function by Factoring and Simplifying

Ex 3: Determine a Limit of a Rational Function by Factoring and Simplifying

Ex 1: Find a Limit by Rationalizing or Factoring

Ex 2: Find a Limit by Rationalizing or Factoring

Determine a Limit Algebraically: Rationalize the Numerator

Determine a Limit of Rational Function: Not Indeterminant Form

Determine Limits of Basic Trigonometric Functions

Determine the Limit of a Trigonometric Function: Factor and Substitution

Determine a Limit of Rational Function: Indeterminant Form (Factor and Simplify)

Determine a Limit of Rational Function: Indeterminant Form (Factor Diff of Cubes)

Ex: Find a Limit Requiring Rationalizing

Determine a Limit Involving a Complex Fraction: LCD (1) - Indeterminant Form

Determine a Limit Involving a Complex Fraction: LCD (2) - Indeterminant Form

Ex: Determine Limits of a Piecewise Defined Function

Determine a Limit of a Rational Function with Negative Exponents Algebraically: LCD

Determine a Limit of a Difference of Rational Functions Algebraically: LCD and Rationalize

Limits of Composite Trigonometric Functions: Direct Substitution

Limits of Exponential Functions with Logarithmic Exponents: Direct Substitution

Limit of a Rational Function: Num/0 (Doesn't Simplify)

Limit of a Rational Function with a Squared Denominator: Num/0 (Doesn't Simplify)

Determine Limits Using Limit Laws (Properties)

Use Limit Laws (Properties) to Determine Limits

Infinite Limits

Determine Infinite Limits of a Rational Function Using a Table and Graph

Determine Infinite Limits of a Rational Function Using a Table and Graph (Squared Denominator)

Limits at Infinity and Special Limits

Limits at Infinity

Limits at Infinity – Additional Examples

Determine Limits at Infinity Numerically Using a Desmos

Ex: Determining Limits at Infinity Graphically

Determine Limits at Infinity and Equations of Horizontal Asymptotes from a Graph

Determine Limits and Equations of Asymptotes from a Graph (Rational)

Determine Limits at Infinity of Rational Functions Using 2 Methods: Degree and Dividing

Determine Limits at Infinity of Rational Functions Using Highest Degree Terms

Limit at Infinity of a Rational Function Using 2 Methods: Degree and Dividing (Constant)

Limit at Infinity of a Rational Function Using 2 Methods: Degree and Dividing: (Infinity)

Limits at Infinity of a Rational Function Using 2 Methods: Degree and Dividing: (Zero)

Limit at Infinity of a Quotient with a Square Root: 3 Methods

Determine Limits at Infinity of Quotients Involving Exponential Terms: 2 Methods (Ex 1)

Determine Limits at Infinity of Quotients Involving Exponential Terms: 2 Methods (Ex 2)

Determine Limits at Infinity of Quotients Involving Exponential Terms: 2 Methods (Ex 3)

Ex: Limits at Infinity of a Polynomial Function

Ex: Limits at Infinity of a Rational Function (DNE)

Ex: Limits at Infinity of a Rational Function (Zero)

Ex: Limits at Infinity of Rational Function (Ratio of Leading Coefficients)

Determine a Limit at Infinity: Rational Function in Factored Form

Ex: Limits at Infinity of a Function Involving a Square Root

Ex: Limits at Infinity of a Function Involving an Exponential Function

Determine Limits at Infinity Involving an Exponential Function: Odd Exponent

Determine Limits at Infinity Involving an Exponential Function: Even Exponent

Determine Limits at Infinity Involving a Natural Log Function

Limits involving Trigonometric Functions

Ex: Find Limits of Composite Function Graphically

Determine Limits at Infinity of a Difference Involving a Square Root

Determine Limits at Infinity of a Sum Involving a Square Root

Determine Special Limits Using a Table and a Graph

Squeeze Theorem and Special Limits

Special Limits in the Form sin(x)/x

Determining Limits Using Special Limits

Continuity Using Limits

Continuity

Intermediate Value Theorem

Ex: Determine Which Rule of Continuity at a Point is Violated

Ex: Continuity at a Point Concept Check

Ex 1: Find the Value of Constant to Make a Piecewise Defined Function Continuous Everywhere

Ex 2: Find the Value of Constant to Make a Piecewise Defined Function Continuous Everywhere

Ex 3: Find the Value of c to Make a Piecewise Defined Function Continuous Everywhere

Determine Values that Make a Piecewise Defined Function Continuous

Determine a Slope to Make a Piecewise Function Continuous Everywhere (Lin/Quad)

Asymptotes: Part 1, Part 2

Limit and Continuity Concept Check with Piecewise Defined Function 1

Limit and Continuity Concept Check with Piecewise Defined Function 2

Determine Where a Piecewise Defined Function Has Discontinuity (Algebra)

Determine Where a Piecewise Defined Function Has Discontinuity (Calculus)

Determine Where a Rational Function with an Exponential Term has Discontinuity

Determine Where a Composite Function has Discontinuity (Ln of Cotangent Squared)

Determine Where and the Type of Discontinuity of a Rational Function

Determine the Interval of Continuity of a Function (square root/quadratic)

Determine the Interval of Continuity of a Function (trig/natural log)

Determine the Interval of Continuity of a Function (quad/trig)

Average Rate of Change

Average Rate of Change

Graphical Approach to Average and Instantaneous Rate of Change

Ex: Determine Average Rate of Change

Ex: Find the Average Rate of Change From a Table - Temperatures

Find an Average Rate of Change from a Table: Movie Receipts

Ex: Find the Average Rate of Change - Miles Per Hour

Ex: Find the Average Rate of Change from a Graph

Ex: Find the Average Rate of Change Given a Function Rule

Ex: Average Rate of Change Application - Hot Air Balloon Function

Ex: Find the Average Rate of Change Given a Function on [2,t]

Ex: Find the Average Rate of Change Given a Function on [3, 3+h]

Estimate a Instantaneous Rate of Change from Average Rates of Change: Water Tank

Function Notation: Building to the Difference Quotient (Quadratic)

Building the Difference Quotient: Quadratic Function

Ex: Use the Slope to Secant Lines to Predict the Slope of a Tangent Line

Estimate the Slope of a Tangent Line from The Slopes of Secant Lines

Ex: Use Average Velocity to Predict Instantaneous Velocity

Estimate Instantaneous Velocity from Average Velocity

Ex: Determine the Intervals for Which the Slope of Tangent Lines is Positive, Negative, and Zero

Ex: Determine the Sign the Slope of a Tangent Line at Point on a Function

Ex: Approximate the Slope of a Tangent Line at a Point on a Function

Ex 1: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 2: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 3: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 4: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex: Determine the Open Intervals Where the First Derivative is Positive or Negative

Ex: Determine the Sign of the First Derivative at a Point on the Graph of a Function

Formal Definition of the Derivative

Introduction to the Derivative

Ex 1: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 2: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 3: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 4: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Estimate a Derivative Function Value from a Table to Values

Ex: Determine the Open Intervals Where the First Derivative is Positive or Negative

Ex: Determine the Sign of the First Derivative at a Point on the Graph of a Function

Comparing the Position and Speed of Two Vehicles from a Graph

Finding Derivatives using the Limit Definition

Find a Function and x-value From Limit Definition of the Derivative

Ex : Determine The Value of a Derivative using the Limit Definition (Quadratic)

Ex : Determine The Value of a Derivative using the Limit Definition (Rational)

Use the Limit Definition of the Derivative to Find a Derivative Function Value

Example 1: Determine a Derivative using The Limit Definition

Example 2: Determine a Derivative using The Limit Definition

Example 3: Determine a Derivative using The Limit Definition

Ex: Determine the Derivative of a Function Using the Limit Definition (ax^2+bx+c)

Use the Limit Definition of the Derivative to Find the Equation of a Tangent Line

Find a Derivative Using The Limit Definition (Quadratic)

Find a Derivative Using The Limit Definition (Cubic)

Application of the Limit Definition of the Derivative (Quadratic Height Function)

Find a Derivative Using The Limit Definition (Rational Function: Linear/Linear)

Find a Derivative Using The Limit Definition (Square Root)

Determine Derivative Function Values Using Desmos

Determine Derivative Function Values Using a Free Online Calculator (MathAS)

Differentiation of Basic Functions and Using the Power Rule

Derivative Flashcards Without The Chain Rule

Derivative Flashcards With The Chain Rule

Differentiation Techniques: Power Rule

Determine Basic Derivatives (Constant, Power Rule)

Determine Basic Derivatives (Radicals)

What is a Tangent Line to a Function?

What is a Normal Line to a Function?

Ex: Derivatives and Derivative Values of a Linear and Constant Function

Ex: Derivative of a Quotient Function By Simplifying

Ex: Find the Equation of a Tangent Line to a Quadratic Function at a Given value of x

Equation of Tangent Line and Normal Line to a Cubic Function

Ex 1: Basic Derivatives Using the Power Rule

Ex: Find the Derivative Function and Derivative Function Value of a Quadratic Function

Ex: Find the Derivative of a Function Containing Radicals

Ex 2: Derivatives Using the Power Rule with Negative and Decimal Exponents

Ex 3: Derivatives Using the Power Rule with Radicals

Ex 4: Derivative Using the Power Rule Involving a Variety of Terms

Ex: Find a Derivative using the Power Rule with Negative Exponents

Determine Basic Derivatives and Derivative Function Values (No Chain)

Find a Derivative Function 2 Ways: Expanding and Product Rule

Ex: Determine Where a Function has Tangent Lines Parallel to a Given Line

Determine the Equation of a Tangent Line and Normal Line to a Function

Determine the Equation of a Tangent Line to a Function

Ex: Find the x-intercept of a Tangent Line

Determine the Points on a Function Where the Tangent Lines are Parallel to a Line

The Derivatives of Sine and Cosine

Ex: Derivative and Derivative Value of Basic Cosine and Sine Functions

Ex: Find the Derivative and Equation of Tangent Line for a Basic Trig Function

Ex: Find a Derivative and Derivative Function Value (Cosine and Cosecant)

Derivatives of Functions in the Form: Radical + Trig

Ex: Find a Derivative of a Function Involving Radicals Using the Power Rule (Rational Exponents)

Ex: Determine the Points Where a Function Has Horizontal Tangent Lines

Ex: Determine the Equation of a Tangent Line to a Function Using the Power Rule

Ex: Determine the Points on a Function When the Tangents Lines Have a Given Slope

Find the Equation of a Tangent Line that is Parallel to a Given Line

Find the Equation of a Normal Line that is Parallel to a Given Line

Determine Points Where a Cubic Function Has Horizontal Tangent Lines

Find the Equation of a Parabola Given 2 Points and the Slope of a Tangent Line

Power Rule of Differentiation App: Rate of Growth of the Radius of an Oil Spill

Determine a 2nd Degree Polynomials Given Function and Derivative Function Values

Find the Equation of a Normal Line and Where it Intersects a Parabola a 2nd Time

Determine the value of the derivative function on the graphing calculator

Determine a Derivative Function Value on the TI84 (Newer Software)

Find the Value of a Derivative Function at a Given Value of x

Applications of the Derivatives Using the Power Rule

Ex 1: Derivative of Trigonometric Functions – Simplify Before Differentiating

Ex: Find the Velocity and Acceleration Function from the Position Function

Interpret the Meaning of a Derivative Function Value (Cost)

Interpret the Meaning of a Derivative Function Value (Population)

Interpret the Meaning of a Derivative Function Value (Temperature)

Find the Rate of Oscillation of a Mass on a Spring: s(t)=6sin(t)

Determine a Function Value and Derivative Value Using Tangent Line

Determine Where a Function is Not Differentiable From a Graph

Ex: Sketch the Graph of a Derivative Function Given the Graph of a Function

Ex 1: Determine the Graph of the Derivative Function Given the Graph of a Quadratic Function

Ex 2: Determine the Graph of the Derivative Function Given the Graph of a Cubic Function

Graph the Derivative Function Given the Graph of a Quadratic Function

Graph the Derivative Function Given the Graph of a Square Root Function

Graph the Derivative Function Given the Graph of a Cubic Function

Graph the Derivative Function Given the Graph of a Degree 4 Function

Identify the Function, Derivative, and 2nd Derivative Given Just the Graphs (1)

Identify the Function, Derivative, and 2nd Derivative Given Just the Graphs (2)

Why is the Derivative of the Area of a Circle Equal to the Circumference?

Why is the Derivative of the Volume of a Sphere Equal to the Surface Area?

Differentiation Using the Product Rule

The Product Rule of Differentiation (Introduction)

Proof: The Product Rule of Differentiation

Ex: Find the Equation of a Tangent Line Using the Product Rule

The Product Rule (old)

Ex: Find a Derivative Using Product Rule (Basic Example)

Ex: Find a Derivative Using Product Rule (Polynomial*Exponential)

Find a Derivative Function Using the Product Rule: Exponential Term

Ex 1: Determine a Derivative Using the Product Rule

Ex 2: Determine a Derivative Using the Product Rule

Ex: Find a Derivative Function Value - Product Rule Concept Check

Ex 1: Determine a Derivative Using the Product Rule Involving a Trig Function

Ex 2: Determine a Derivative Using the Product Rule Involving a Trig Function

Derivatives Using the Product Rule in the Form: Trig * Trig

Derivatives Using the Product Rule in the Form: Radical * Trig

Ex: Determine the Equation of a Tangent Line Using the Product Rule

Ex: Find a Derivative Using the Product Rule (Linear*Trig) and Find Equation of Tangent Line

Ex: Find a Derivative and Equation of Tangent Line Using Product and Chain Rule (Exp*Trig)

Ex: Find a Derivative Function and Derivative Value Using the Product Rule (3 products)

Ex 1: Derivative of Trigonometric Functions – Simplify Before Differentiating

Ex 2: Derivative of Trigonometric Functions Using Product Rule – Simplify Before Differentiating

Derivative of a Product Involving Inverse Tangent: y=(4sin(x)+8cos(x))arctan(x)

Derivative Functions: Product Rule Within Product Rule

Differentiation Using the Quotient Rule

The Quotient Rule

Ex: Use the Quotient Rule to Find the Derivative and Derivative Value (Basic)

Ex 1: Quotient Rule or Power Rule to Find a Derivative (Comparison)

Ex 2: Quotient Rule or Power Rule to Find a Derivative (Comparison)

The Product and Quotient Rule With Trigonometric Functions

Ex 1: Determine a Derivative Using the Quotient Rule

Ex 2: Determine a Derivative Using the Quotient Rule

Ex 3: Determine a Derivative Using the Quotient Rule

Determine a Derivative using The Quotient Rule: Sine over Polynomial

Derivative Functions: Quotient Rule with Trigonometric Functions

Determine a Derivative using The Quotient Rule: Sine over Polynomial

Determine a Derivative using The Quotient Rule: Form of Cot Over Csc

Concept Check: Product and Quotient Rule of Differentiation

Find Derivative Function Values Using Sum, Product, and Quotient Rules (Function Notation Only)

Find the First and Second Derivative Using the Quotient Rule (Mono Over Squared Binomial)

Find a Derivative using the Quotient Rule (Exponential Over Monomial)

The Derivative of a Rational Function Using the Quotient Rule (Powers of Binomials)

Ex: Find a Derivative Function Value Using the Quotient Rule and by Interpreting a Graph

Ex: Find a Derivative and Derivative Function Value Using the Quotient Rule (square roots)

Ex: Find a Derivative and Derivative Function Value Using the Quotient Rule (linear/trig)

Ex: Find a Derivative and Using the Quotient Rule (trig/poly)

Ex: Find the X-values Where a Function has Derivative Function Value (Quotient Rule)

Ex: Determine the Slope of a Tangent Line Using the Quotient Rule

Ex: Derivative with The Quotient Rule Involving Trig Functions - Equation of Tangent Line

Ex: Derivative and Derivative Function Value Using the Quotient Rule (Tangent)

Ex: Determine the Equation of a Tangent Line to Using the Quotient Rule Involving a Trig Function

Ex 1: Determine a Derivative Using the Quotient Rule Involving a Trig Function

Ex 2: Determine a Derivative Using the Quotient Rule Involving a Trig Function

Average Revenue, Cost, Profit Functions and their Derivatives

Differentiation Using the Chain Rule

The Chain Rule: Part 1, Part 2

Find a Derivative Function 2 Ways: Expanding and Chain Rule

The Chain Rule: Derivative of a Function that is a Cube of a Polynomial

The Chain Rule with Transcendental Functions

Ex 1: Chain Rule Concept Check

Ex 2: Power Rule with Chain Rule Concept Check

Ex 3: Power Rule with Chain Rule Concept Check

Ex 4: Power Rule with Chain Rule Concept Check

Determine a Derivative Function Using the Chain Rule: e to a Product

Determine a Derivative Function Using the Chain Rule (Ln of Linear Function)

Determine a Derivative Function Using the Chain Rule (Cosine of Ln)

Determine a Derivative Function Using the Chain Rule (Ln of a Sum)

Determine a Derivative Function Using the Chain Rule (Cosine)

Ex: Derivatives Using the Chain Rule - Quadratic Raised to a Power

Ex: Derivatives Using the Chain Rule - Negative Exponent

Ex 1: Determine a Derivative Using the Chain Rule

Ex 2: Determine a Derivative Using the Chain Rule

Ex 3: Determine a Derivative Using the Chain Rule

Ex 4: Determine a Derivative Using the Chain Rule Involving an Exponential Function

Ex 5: Determine a Derivatives using The Chain Rule Involving Trig Functions

Ex: Derivatives Using the Chain Rule Involving a Trigonometric Functions

The Chain Rule: Determine the First and Second Derivative of y=cos(2x^5)

Ex: Derivatives Using the Chain Rule Involving an Exponential Function with Base e

The Chain Rule: Equation of a Tangent Line to y=2/3sin(sin(x))

The Chain Rule: Derivative and Derivative Function Value of a Power of Sine

The Chain Rule: Derivative and Derivative Function Value of Sine of a Power of x

Ex: Derivative using the Product Rule and Chain Rule – Product of Polynomials to Powers

Ex 1: Determine a Derivative Using the Chain Rule and Product Rule

Ex 2: Determine a Derivative Using the Chain Rule and Product Rule Involving a Radical

Ex 3: Determine a Derivative Using the Chain Rule and Product Rule With a Trig Function

The Chain Rule: Derivatives of Composite Trigonometric Functions

The Chain Rule: The Derivative of a Power of a Composite Trigonometric Functions

The Chain Rule: The Derivative of a Trigonometric Function of a Quotient

The Derivative of an Exponential Function using the Chain and Product Rules

Find a Derivative Using the Product Rule (Exponential Functions)

Find the Derivative of the Square Root of a Log Function

Ex: Determine a Derivative Using the Chain Rule and Quotient Rule

Determine a Derivative Function Using the Chain Rule: Comp of Three Functions

Ex: Derivative Using the Chain Rule Twice - Trig Function Raised to Power

Ex: Derivative Using the Chain Rule Twice - Exponential and Trig Functions

The Chain Rule: Derivative of a Function that is a Composition of 3 Trigonometric Functions

Derivative of a Composite Inverse Tangent Function: y=6arctan(8sin(3x))

Chain Rule Concept Check: Derivative Function Value Using Function Notation

Derivatives with Function Notation: Product, Quotient, and Chain Rule Concept Check

Chain Rule Concept Check: Derivative Function Value of a Radical Function Using Function Notation

Differentiation of Exponential Functions

Graphing Exponential Functions

Introduction to Derivatives of Logarithmic and Exponential Functions (no chain rule)

Derivative of an Function with an Exponential Term (Base e) and Slope of Tangent Line (no chain rule)

Derivatives of Exponential Functions with base e

Ex 1: Derivatives Involving the Exponential Function with Base e

Ex 2: Derivatives Involving the Exponential Function with Base e and the Product Rule

Ex 3: Derivatives Involving the Exponential Function with Base e and the Power Rule

Ex 4: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 5A: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 5B: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 1: Derivatives of Exponential Functions

Ex 2: Derivatives of Exponential Functions With Chain Rule

Ex 3: Derivatives of Exponential Functions with the Product Rule

Ex 4: Derivatives of Exponential Functions with the Quotient Rule

Ex: Derivatives Using the Chain Rule Involving an Exponential Function with Base e

The Derivative of an Exponential Function using the Chain and Product Rules

Find a Derivative Using the Product Rule (Exponential Functions)

Ex: Find the Equation of a Tangent Line at a Given Point – Linear and Exponential Function

Ex: Application of the Derivative of an Exponential Function (Rate of Depreciation)

Derivative App: Rate of Growth of People Infected by Flu y=ae^(kt)

Differentiation of Hyperbolic Functions

Introduction to Hyperbolic Functions

Prove a Property of Hyperbolic Functions: (sinh(x))^2 - (cosh(x))^2 = 1

Prove a Property of Hyperbolic Functions: (tanh(x))^2 + (sech(x))^2 = 1

Prove a Property of Hyperbolic Functions: sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y)

Prove a Property of Hyperbolic Functions: (sinh(x))^2=(-1+cosh(2x))/2

Ex 1: Derivative of a Hyperbolic Function

Ex 2: Derivatives of Hyperbolic Functions with the Chain Rule

Ex 3: Derivative of a Hyperbolic Function Using the Product Rule

Ex 4: Derivative of a Hyperbolic Function Using the Quotient Rule

Ex 5: Derivatives of Hyperbolic Functions with the Chain Rule Twice

Ex 1: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Ex 2: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Ex 3: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Differentiation of Logarithmic Functions

Logarithms

Derivatives of Logarithmic Functions

Introduction to Derivatives of Logarithmic and Exponential Functions (no chain rule)

Derivative of an Logarithmic Function and Slope of Tangent Line (log(x) and ln(x), no chain rule)

Derivative of Natural Log of a Trig Function: y=ln(-4csc(x)) and y=ln(3sec(x))

Ex 1: Derivatives of the Natural Log Function

Ex 2: Derivatives of the Natural Log Function with the Chain Rule

Ex 3: Derivatives of the Natural Log Function with the Chain Rule

Ex 4: Derivatives of the Natural Log Function with the Chain Rule

Derivative of Natural Log of a Power of a Trig Function: y=ln(2sin^6(x))

Ex 5: Derivatives of the Natural Log Function with the Product Rule

The Derivative of A Natural Log Function Using Properties of Logarithms

Derivative of a Natural Log Function Using Log Properties

Ex 6: Derivatives of the Natural Log Function using Log Properties

Ex 7: Derivatives of the Natural Log Function using Log Properties

Ex 8: Derivatives of the Natural Log Function using Log Properties

Ex 9: The derivative of f(x) = ln(ln(5x))

Derivatives of a^x and logax

Ex 1: Derivative of the Log Function, not base e

Ex 2: Derivative of the Log Function using the Product Rule

Derivative of the Log of a Product Involving an Exponential: y=log_3(2xe^x+1)

The Derivative of a Logarithmic Function Using the Product Rule

Derivative of a Product Involving Natural Log: y=6x^2*ln(3x)-4x^3

Derivative of a Composite Function: y=sin(2ln(x)) and y=cos(-3ln(x))

Derivative and Slope of a Tangent Line of a Natural Log Function: y=ln(x+sqrt(x^2+32))

The First and Second Derivative of a Quotient Involving Natural Log

Given a Derivative Function Value Find a Missing Coefficient of the Function

Logarithmic Differentiation

Logarithmic Differentiation

Logarithmic Differentiation: sin(x) to the power of x

Logarithmic Differentiation: 2+x to the power of 2/x

Find a Derivative Using Logarithmic Differentiation

The Derivative of a Rational Function Using the Log Differentiation (Powers of Binomials)

Ex: Logarithmic Differentiation

Logarithmic Differentiation of a Quotient

Logarithmic Differentiation: x^(ax)

Ex 1: Logarithmic Differentiation

Ex 2: Logarithmic Differentiation and Slope of a Tangent Line

Ex 3: Logarithmic Differentiation and Slope of a Tangent Line

Logarithmic Differentiation: x^(3sin(x))

Logarithmic Differentiation: (sqrt(x))^(7x)

Logarithmic Differentiation: (cos(3x))^(4x)

Logarithmic Differentiation: 8x^(ln(x))

Differentiation of Inverse Trigonometric Functions

Ex: Find an Inverse Derivative Function Value (Cubic)

Ex: Find an Inverse Derivative Function Value (Cubic + Rational)

Ex: Find an Inverse Derivative Function Value (Sine)

Ex: Find an Inverse Derivative Function Value (Square Root)

The Derivatives of the Inverse Trigonometric Functions

Ex 1: Derivatives of Inverse Trig Functions

Ex 2: Derivatives of Inverse Trig Functions

Ex 3: Derivatives of Inverse Trig Functions

Derivative of Arcsine and Arccosine with the Chain Rule

Derivative of Arctangent and Arcsecant with the Chain Rule

Find the Derivative of a Composite Function Involving Natural Log and Hyperbolic Functions

Determine the Rate of Change of an Angle of Elevation Using Arcsine: Pole Wires

Determine the Rate of Change of an Angle of Elevation Using Arctangent: Building Shadow

Higher Order Differentiation

Higher-Order Derivatives: Part 1, Part 2

Determine a First and Second Derivative: Rational Exponent

Higher Order Derivatives of Transcendental Functions

Ex 1: Determine Higher Order Derivatives

Ex 2: Determine Higher Order Derivatives

Ex 3: Determine Higher Order Derivatives

Ex 4: Determine Higher Order Derivatives Requiring the Chain Rule

Ex 5: Determine Higher Order Derivatives Requiring the Product Rule and Chain Rule

Ex 6: Determine Higher Order Derivatives Requiring the Quotient Rule

Ex: Find Higher Order Derivatives of Sine

Ex: Higher Order Derivatives Using the Product Rule

Ex 1: First and Second Derivatives Using the Chain Rule - f(x)=tan(2x)

Ex 2: First and Second Derivatives Using the Chain Rule - f(x)=ln(cos(x))

First and Second Derivative Functions Using the Product Rule: Exp*Cosine

Ex: Find the First and Second Derivative Functions and Function Value (Exponential and Polynomial)

The Second Derivative of 5ln(sec(x)+tan(x))

Distance Traveled from the Graph of a Position Graph

Position Function Applications: Velocity, Up/Down Movement, Acceleration

Velocity Function Graph Application: Height Up/Down, Acceleration Incr/Decr

Ex: Determine the Velocity Function and Acceleration Function from the Position Function

Applications of Differentiation – Relative Extrema

Ex: Find the Critical Numbers of a Cubic Function

Determine the Critical Numbers of a Rational Function (1)

Determine the Critical Numbers of a Rational Function (2): Irrational

Determine the Critical Numbers of a Function Involving Difference of Radicals

Determine the Critical Numbers of a Function Involving The Product of a Radical and Binomial

Determine the Critical Numbers of a Function Involving Natural Log

Increasing and Decreasing Functions

Ex: Determine Increasing or Decreasing Intervals of a Function

Ex 1: Determine the Intervals for Which a Function is Increasing and Decreasing

Ex 2: Determine the Intervals for Which a Function is Increasing and Decreasing

Using the Graph of the Derivative to Determine Where a Function is Increasing or Decreasing (Cubic)

Determine the Critical Number and Relative Extrema of a Quadratic Function

Determine the Critical Numbers and Relative Extrema of a Degree 5 Polynomial (Quad Formula)

Ex: Determine Increasing/Decreasing Intervals and Relative Extrema

Determine Relative Extrema Using the First Derivative Test

Ex: Determine Increasing/Decreasing Intervals and Relative Extrema (Product Rule with Exponential)

Ex: Determine Increasing/Decreasing Intervals and Absolute Extrema (Product Rule)

Ex: Find the Intervals Incr/Decr and Relative Extrema Using the First Derivative

Ex: Find the Intervals Incr/Decr and Relative Extrema (Quad Formula Used)

Determine where a trig function is increasing/decreasing and relative extrema

Determine Local (Relative) Extrema of a Polynomial Function Using the TI-84

Determine Local (Relative) Extrema of a Polynomial Function Using Desmos

Ex 1: First Derivative Concept - Given Information about the First Derivative, Describe the Function

Ex 2: First Derivative Concept - Given Information about the First Derivative, Describe the Function

Ex 1: Interpret the Graph of the First Derivative Function – Degree 2

Ex 2: Interpret the Graph of the First Derivative Function - Degree 3

The First Derivative Test to Find Relative Extrema

Ex: Critical Numbers / Relative Extrema / First Derivative Test

Determining Relative Extrema on the Graphing Calculator

Ex 1: Determine Relative Extrema Using The First Derivative Test

Ex 2: Determine Relative Extrema Using The First Derivative Test Involving a Rational Function

Ex 3: Determine Relative Extrema Using The First Derivative Test Involving a Trig Function

Ex 1: Sketch a Graph Given Information About a Function's First Derivative

Ex 2: Sketch a Graph Given Information About a Function's First Derivative

Determine the Maximum Height Given a Height Function Using the First Derivative

Minimize a Cost Function Using the First Derivative Test

Finding Max and Mins Applications: Part 1, Part 2

Derivative Concept Check: Determine Smallest Possible Function Value at the Endpoint of an Interval

Derivative Concept Check: Determine the Max and Min of f(b)-f(a) Given Interval for Derivative Values

Use the Graph of the Derivative Function to Determine Incr/Decr Intervals and Location of Relative Extrema

Business Applications of Differentiation and Relative Extrema

Determine the Marginal Supply Function from the Supply Function (Chain Rule App)

Ex: Optimization - Maximized a Crop Yield (Calculus Methods)

Ex: Profit Function Applications – Average Profit, Marginal Profit, Max Profit

Ex: Profit Function Application - Maximize Profit

Elasticity of Demand: Part 1, Part 2

Elasticity of Demand: D(x)=sqrt(300-0.5x^3)

Ex: Elasticity of Demand - Quadratic Demand Function

Determine Elasticity of Demand and Unit Elasticity Price (Linear Demand)

Exponential Growth Models Part 1, Part 2

Exponential Decay Models: Part 1, Part 2

Marginals

Ex: Marginals and Marginal Profit

Ex: Marginals and Marginal Average Cost

Applications of Differentiation – Concavity

Determining the Concavity of a Function

Concavity of Transcendental Functions (Additional Examples)

Ex: Given the First Derivative, Describe the Function (Incr/Decr/CCU/CCD)

Ex: Determine Concavity and Points of Inflection

Ex: Concavity of a Degree 5 Polynomial - Irrational Critical Numbers

Determine the Concavity and Points of Inflection: y=xe^(bx)

Ex: Determine Concavity and Absolute Extrema (Product and Quotient Rule)

Ex: Determine Increasing/Decreasing/Concavity Intervals of a Function

Ex: Determine Increasing/Decreasing/Concavity Intervals of a Rational Function

Ex: Determine Concavity and Points of Inflection - f(x)=x^2*e^(4x)

Ex: Find the Intervals a Function is Increasing/Decreasing/Concave Up or Down - Rational Exponent

Ex: Determine Increasing / Decreasing / Concavity by Analyzing the Graph of a Function

The Second Derivative Test to Determine Relative Extrema

The Second Derivative Test to Determine Relative Extrema: y=-ax^5+bx^3

The Second Derivative Test to Determine Relative Extrema: y=xe^(bx)

Ex 1: The Second Derivative Test to Determine Relative Extrema

Ex 2: The Second Derivative Test to Determine Relative Extrema

Ex: Critical Numbers / Relative Extrema / Second Derivative Test

The Second Derivative Test using Transcendental Functions

Example: Increasing/Decreasing / Concavity / Relative Extrema / Points of Inflection

Ex 1: Sketch a Function Given Information about Concavity

Ex 2: Sketch a Function Given Information about Concavity

Ex: Determine the Sign of f(x), f'(x), and f''(x) Given a Point on a Graph

Ex 1: Intervals for Which the First and Second Derivative Are Positive and Negative Given a Graph

Ex 2: Intervals for Which the First and Second Derivative Are Positive and Negative Given a Graph

Use the Graph of the Derivative Function to Determine Concavity and Location of Points of Inflection

Concept Check: Determine the Sign of 1st and 2nd Derivatives for Real Life Situations

Applications of Differentiation – Maximum/Minimum/Optimization Problems

Ex 1: Max / Min Application Problem - Derivative Application

Ex 2: Max / Min Application Problem - Derivative Application

Ex 3: Max / Min Application Problem - Derivative Application

Ex: Optimization - Maximized a Crop Yield (Calculus Methods)

Ex: Derivative Application - Minimize Cost

Ex: Derivative Application - Maximize Profit

Ex: Optimization - Maximum Area of a Rectangle Inscribed by a Parabola

Ex: Optimization - Minimize the Surface Area of a Box with a Given Volume

Ex: Optimization - Minimize the Cost to Make a Can with a Fixed Volume

Optimization: Determine The Nitrogen Level that Maximized a Crop Yield

Optimization: Lifeguard Problem - Find Run Distance Minimize Rescue Time with Run and Swim

Optimization: Maximize Suitcase Volume Given Sum of Length, Width, Height (Airline)

Optimization: Maximum Area of a Rectangle Bounded by a Parabola and X-axis

Optimization: Maximize the Vertical Distance Between a Line and Parabola

Ex: Derivative Application - Maximize Profit

Ex: Derivative Application: Maximize Area

Ex: Derivative Application - Minimize the Cost of a Fenced Area

Optimization - Maximize the Area of a Norman Window

Ex: Find the Average Cost Function and Minimize the Average Cost

Ex 1: Cost Function Applications - Marginal Cost, Average Cost, Minimum Average Cost

Ex 2: Cost Function Applications - Marginal Cost, Average Cost, Minimum Average Cost

Ex: Find a Demand Function and a Rebate Amount to Maximize Revenue and Profit

Ex: Given the Cost and Demand Functions, Maximize Profit

Animation: The graphs of f(x), f’(x), f’’(x)

Absolute Extrema

Determine the Relative and Absolute Extrema from The Graph of a Function

Determine the Relative and Absolute Extrema from The Graph of a Common Log Function

Determine the Relative and Absolute Extrema from The Graph of Trigonometric Function

Determine the Relative and Absolute Extrema from The Graph of a Piecewise Function

Absolute Extrema

Absolute Extrema on a Close Interval: Cubic Function

Absolute Extrema on a Close Interval: Quartic Function

Absolute and Relative Extrema From a Graph (Closed Interval)

Determine Absolute Extrema on a Closed Interval: Natural Log Function

Absolute and Relative Extrema From a Graph (Open Interval)

Absolute Extrema of Transcendental Functions

Ex 1: Absolute Extrema on an Closed Interval

Ex 2: Absolute Extrema on an Open Interval

Ex: Absolute Extrema of a Quadratic Function on a Closed Interval

Ex: Absolute Extrema of a Trigonometric Function on a Closed Interval

Ex: Determine Increasing/Decreasing Intervals and Absolute Extrema (Product Rule)

Differentials and Tangent Line Approximations

Introduction to Differentials

Tangent Line Approximation Given a Function and Derivative Function Value

Make a Tangent Line Approximation for a Square Root Function Value

Ex: Use a Tangent Line to Approximate a Square Root Value

Determine a Linear Approximation for f(x)=sin(3x)

Ex: Use a Tangent Line to Approximate a Quotient

Ex: Use a Tangent Line to Approximate a Cube Root Function Value – Chain Rule

Tangent Line Approximation: Estimate Function Values Given Point and Derivative Function Value

Differentials

Ex 1: Determine Differential y (dy)

Ex 2: Differentials: Determine dy given x and dx

Ex: Differentials to Approximate Propagated Error and Relative Error

Ex: Using Differentials to Approximate the Value of a Cube Root.

Ex: Differentials: Compare delta y and dy

Ex: Find dy Given a Tangent Function - Requires the Chain Rule

Determine Absolute Error and Percent Error

Ex: Differentials - Approximate Delta y Using dy Using a Sine Function and Find Error Percent

Ex: Use Differentials to Approximate Possible Error for the Surface Area of a Sphere

Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem

Proof of Rolle's Theorem

Ex 1: Rolle's Theorem

Ex 2: Rolle's Theorem with Product Rule

The Mean Value Theorem

Proof of the Mean Value Theorem

Can You Be Ticketed for Speeding? (Mean Value Theorem)

Ex 1: Mean Value Theorem – Quadratic Function

Ex 2: Mean Value Theorem – Cubic Function

Ex 3: Mean Value Theorem – Rational Function

Ex 4: Mean Value Theorem – Quadratic Fomula Needed

Implicit Differentiation

Introduction to Basic Implicit Differentiation

Implicit Differentiation

Implicit Differentiation: Determine the Equation of the Tangent Line: x^(1/2)+y^(1/2)=7

Implicit Differentiation of Equations containing Transcendental Functions

Implicit Differentiation: e^(2xy)+y^4+x^3=1

Ex 1: Implicit Differentiation

Ex 2: Implicit Differentiation Using the Product Rule

Ex 3: Implicit Differentiation Using the Product Rule and Factoring

Ex 4: Implicit Differentiation Involving a Trig Function

Ex: Implicit Differentiation - Equation of Tangent Line

Ex: Implicit Differentiation Involving a Trig Function

Ex: Implicit Differentiation to Determine a Second Derivative

Ex: Perform Implicit Differentiation and Find the Equation of a Tangent Line

Ex: Find dy/dx Using Implicit Differentation and the Product Rule - e^(2xy)=y^n

Ex: Find dy/dx Using Implicit Differentation and the Product Rule - ax-bxy-cy^n=d

Implicit Differentiation: x^n(x+y)=y^m(ax-y)

Implicit Differentiation: ysin(xy)=y^6-5

Implicit Differentiation: Equation of Tangent Line xy+cos(xy)-4x=-7

Implicit Differentiation with Trigonometric Functions To Determine a Tangent and Normal Line

Implicit Differentiation: tan(x-y)+y/(5+2x^2)

Implicit Differentiation: ysqrt(x+9y)=2xy-3 (Challenging)

Implicit Differentiation Using Function Notation: Concept Check

arImplicit Differentiation: Find dh/dr Using the Volume of a Cone Formula

Related Rates

Related Rates

Related Rates: Determine dy/dt Given a Function of x (Basic Example)

Ex 1: Related Rates: Determine the Rate of Change of Profit with Respect to Time

Ex 2: Related Rates: Determine the Rate of Change of the Area of a Circle With Respect to Time

Ex 3: Related Rates: Determine the Rate of Change of Volume with Respect to Time

Solve a Related Rates Problem Using the Product Rule

Related Rates Applicate: Leaking Conical Tank

Ex 4: Related Rates: Ladder Problem

Ex: Related Rates - Area of Triangle

Ex: Related Rates - Right Circular Cone

Ex: Related Rates - Rotating Light Projecting on a Wall

Ex: Related Rates - Volume of a Melting Snowball

Ex: Related Rates - Air Volume and Pressure

Ex: Related Rates Problem – Rate of Change of a Shadow from a Light Pole

Ex 2: Related Rates Problem -- Rate of Change of a Shadow from a Light Pole

Ex: Related Rates Problem -- Rate of Change of Distance Between Ships

Ex: Related Rates - Find the Rate of Change of Revenue

Ex: Related Rates - Find the Rate of Change of Revenue (Quotient Rule)

Related Rates: Determine The Rate of Change of the Height of Water in a Leaking Cylinder

Related Rates: Determine The Rate of Change of Angle of Elevation Watching a Bird

Related Rates: Determine The Rate of Change of Angle of Elevation of Sliding Ladder (No Ladder Length)

Related Rates: Determine The Rate of Change of Top of Sliding Ladder

Related Rates: Determine The Rate of Change of the Area Formed by Sliding Ladder

Newton’s Method and L’Hopital’s Rule

Newton’s Method

Ex: Newton’s Method to Approximate Zeros – 2 Iterations

Newton's Method: Perform One Iteration Using Desmos (y=(ln x)/x)

Newton's Method: Approximate One Solution to a Cubic Equation (Table Using Desmos)

Newton's Method: Find the Location of a Maximum Value Using MOER App (y=e^(x/3)-x^2)

Newton's Method: Find the Location of a Maximum Value Using a Table In Desmos (y=sin(x)-2x^2)

Newton's Method: Solve a Quadratic Equation Not Equal to Zero Using MOER App (x^2=c)

L’Hopital’s Rule: Part 1, Part 2

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 1)

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 2)

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 3)

L'Hopital's Rule - Justification Using Tangent Lines (Form 0/0)

Partial Proof of L'Hopital's Rule (Only Form 0/0)

Ex 1: L'Hopitals Rule Involving Trig Functions

Ex 2: L'Hopitals Rule Involving Trig Functions

Ex 3: L'Hopitals Rule Involving Exponential Functions

Ex: Use L'Hopital's Rule to Determine a Limit Approaching Infinity

Ex: Use L'Hopital's Rule to Determine a Limit Approaching Zero

Ex 1: Use L'Hopital's Rule to Determine a Limit Approaching Zero with Trig Function

Ex 2: Use L'Hopital's Rule to Determine a Limit Approaching Zero with Trig Function

Use L'Hopital's Rule to Determine a Limits: (ae^x-b)/(sin(c)) and sin(ax)/sin(bx)

Use L'Hopital's Rule to Determine a Limit: (axln(bx))

Use L'Hopital's Rule to Determine a Limit: (lnx)/(5^(ln x)-x)

Use L'Hopital's Rule to Determine a Limit: (4/x-4/sin(x))

Use L'Hopital's Rule to Determine a Limit: (sqrt(1+bx)-sqrt(1-cx))/(2x)

Use L'Hopital's Rule to Determine a Limit: (e^(ax)-b-cx)/(dx^2)

Use L'Hopital's Rule to Determine a Limit: (ax-sin(x))/(bx-tan(x))

Use L'Hopital's Rule to Determine a Limit: (x7^x)/(7^x-1)

Use L'Hopital's Rule to Determine a Limit: (ln x)^2/(ax^2)

Proofs

The Squeeze Theorem

Prove the Limit as x Approaches 0 of sin(x)/x

Prove the Limit as x Approaches 0 of (1-cos(x))/x

Prove the Limit as x Approaches 0 of (e^x-1)/x

Prove the Derivative of a Constant: d/dx[c]

Proof - the Derivative of a Constant Times a Function: d/dx[cf(x)]

Proof - the Derivative of Sum and Difference of Functions: d/dx[f(x)+g(x)]

Proof - The Derivative of Sine: d/dx[sin(x)]

Proof - The Derivative of Cosine: d/dx[cos(x)]

Proof - The Power Rule of Differentiation

Proof - The Product Rule of Differentiation

Proof - The Quotient Rule of Differentiation

Proof - The Chain Rule of Differentiation

Proof - The Derivative of f(x) = e^x: d/dx[e^x]=e^x (Limit Definition)

Proof - The Derivative of f(x) = e^x: d/dx[e^x]=e^x (Implicit Differentiation)

Proof - The Derivative of f(x)=ln(x): d/dx[ln(x)]=1/x (Implicit Diff)

Proof - The Derivative of f(x)=log_a(x): d/dx[log_a(x)]=1/((ln a)x)

Proof - The Derivative of f(x)=a^x: d/dx[a^x]=(ln a)a^x (Definition)

Proof - The Derivative of f(x)=a^x: d/dx[a^x]=(ln a)a^x (Using Logs)

Proof - The Derivative of Tangent: d/dx[tan(x)]

Proof - The Derivative of Cotangent: d/dx[cot(x)]

Proof - The Derivative of Secant: d/dx[sec(x)]

Proof - The Derivative of Cosecant d/dx[csc(x)]

Proof - The Derivative of f(x)=arcsin(x): d/dx[arcsin(x)]

Proof - The Derivative of f(x)=arccos(x): d/dx[arccos(x)]

Proof - The Derivative of f(x)=arctan(x): d/dx[arctan(x)]

Proof - The Derivative of f(x)=arccot(x): d/dx[arccot(x)]

Proof - The Derivative of f(x)=arccsc(x): d/dx[arccsc(x)]

Proof - The Derivative of f(x)=arcsec(x): d/dx[arcsec(x)]

Proof of Rolle's Theorem

The Mean Value Theorem

Proof of the Mean Value Theorem

L'Hopital's Rule - Justification Using Tangent Lines (Form 0/0)

Partial Proof of L'Hopital's Rule (Only Form 0/0)