What is Calculus?

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

Use a Position Function to Determine the Velocity and Acceleration Functions and When Object is Speeding Up or Slowing

Use the Graphs of the Velocity and Acceleration Functions to Determine if an Object is Speeding Up or Slowing Down

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:  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)

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