London Bridge, Lake Havasu City, Arizona

Slides:

Advertisements

Similar presentations

3.3 Differentiation Rules

Advertisements

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. 2.3 Derivatives of Trigonometric.

3.5 Derivatives of Trig Functions. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine.

1 The Product and Quotient Rules and Higher Order Derivatives Section 2.3.

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

2-1 The Derivative and the Tangent Line Problem 2-2 Basic Differentiation Rules and Rates of Change 2-3 Product/Quotient Rule and Higher-Order Derivatives.

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 2.3 Product and Quotient Rules.

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 AP Calculus AB/BC 3.3 Differentiation Rules,

Product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 3.4 Derivatives of Trig Functions.

Bell-ringer 11/2/09 Suppose that functions f and g and their derivatives with respect to x have the following values at x=0 and x=1. 1.Evaluate the derivative.

Hypatia of Alexandria 370 – 415 Hypatia of Alexandria 370 – 415 Hypatia was an Alexandrine Greek Neoplatonist philosopher in Egypt who was one of the earliest.

The Second Derivative Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Photo by Vickie Kelly, 2003 Arches.

Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College Photo by Vickie Kelly, 2001 London Bridge, Lake Havasu City,

3.3 Differentiation Rules Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

2.3 Basic Differentiation Formulas

Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.5 Implicit Differentiation

2.3 The Product and Quotient Rules (Part 1)

London Bridge, Lake Havasu City, Arizona

Implicit Differentiation

Derivatives of Trig Functions

2.3 Basic Differentiation Formulas

3.3 Rules for Differentiation

5.6 Inverse Trig Functions and Differentiation

3.2: Rules for Differentiation

Differentiation Rules

3.3 Differentiation Rules

2.1 The Derivative and the Tangent Line Problem (Part 2)

3.4 Derivatives of Trig Functions

London Bridge, Lake Havasu City, Arizona

Derivatives of Trig Functions

Derivatives of Trig Functions

Derivatives of Trig Functions

Mark Twain’s Boyhood Home

Mark Twain’s Boyhood Home

5.4: Exponential Functions: Differentiation and Integration

3.3 Differentiation Rules

Black Canyon of the Gunnison National Park, Colorado

5.3 Inverse Function (part 2)

5.2 (Part II): The Natural Logarithmic Function and Integration

5.2 (Part I): The Natural Logarithmic Function and Integration

Black Canyon of the Gunnison National Park, Colorado

3.3 Differentiation Rules

Rules for Differentiation

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

5.7 Inverse Trig Functions and Integration (part 2)

5.7 Inverse Trig Functions and Integration (part 1)

3.4 Derivatives of Trig Functions

3.3 Rules for Differentiation

2.4 The Chain Rule (Part 2) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

4.2 Area Greenfield Village, Michigan

5.2 (Part II): The Natural Logarithmic Function and Integration

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

5.1 (Part I): The Natural Logarithmic Function and Differentiation

U.S.S. Alabama 2.4 Chain Rule Mobile, Alabama

The Chain Rule Section 3.6b.

3.3 Differentiation Rules

3.7 Implicit Differentiation

5.2 (Part I): The Natural Logarithmic Function and Integration

3.5 Derivatives of Trig Functions

Parametric Functions 10.1 Greg Kelly, Hanford High School, Richland, Washington.

3.5 Derivatives of Trig Functions

3.3 Differentiation Rules

2.1 The Derivative and the Tangent Line Problem (Part 2)

2.5 Basic Differentiation Properties

5.4: Exponential Functions: Differentiation and Integration

5.3 Inverse Function (part 2)

Presentation transcript:

2.3 The Product and Quotient Rules and Higher-Order Derivatives (Part 2) London Bridge, Lake Havasu City, Arizona Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2001

Objectives Find the derivative of a trigonometric function. Find a higher-order derivative of a function.

Consider the function slope We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.

We can do the same thing for slope The resulting curve is a sine curve that has been reflected about the x-axis.

We can find the derivative of tangent x by using the quotient rule.

Derivatives of the remaining trig functions can be determined the same way.

Example: Find the derivative

Example: Find the derivative

Find the derivative:

Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative. is the fourth derivative.

Higher Order Derivatives: See notation on page 125: y f(x) y' f '(x) dy/dx y'' f ''(x) d2y/dx2 y''' f '''(x) d3y/dx3 y (4) f (4)(x) d4y/dx4

Homework 2.3 (page 126) #39-67 odd 75, 79, #93-107 odd, # 117