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The Product Rule Today, students will analyze a proof of the Product Rule and learn when to apply the Product Rule

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The Product Rule. Today, students will analyze a proof of the Product Rule and learn when to apply the Product Rule. Today, you are going to find the derivative of functions of the form j(x)=f(x) · g(x). Some of these are easy to do using algebra to expand ahead of time:. - PowerPoint PPT Presentation

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Page 1: The Product Rule

The Product Rule

Today students will analyze a proof of the Product Rule and learn when to

apply the Product Rule

Today you are going to find the derivative of functions of the form j(x)=f(x)g(x) Some of these are easy to do using algebra to expand ahead of

time

2

Find the derivative of

a 3

b 10 5

c 1

j x x x

j x x x

j x x

Do you notice any patterns

Try some more difficult cases

2 3

Find the derivative of

a 2 1 5

b 5 1 10

c 3 1 2

j x x x

j x x x

j x x x

Do you notice any patterns

4

Letrsquos consider a function thatrsquos not easy to expand

sinj x x x

It would be nice to develop a method to differentiate the product of two functions

Explain how the function above can be modeled as j(x)=f(x)g(x)

5

The Derivative of a ProductWhen Hana looked at the function j(x)=f(x)g(x) she immediately thought of a rectangle whose length and width were f(x) and g(x) and whose area was j(x) Hana would like to find jrsquo(x) by using the definition of the derivative as a limit

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 2: The Product Rule

Today you are going to find the derivative of functions of the form j(x)=f(x)g(x) Some of these are easy to do using algebra to expand ahead of

time

2

Find the derivative of

a 3

b 10 5

c 1

j x x x

j x x x

j x x

Do you notice any patterns

Try some more difficult cases

2 3

Find the derivative of

a 2 1 5

b 5 1 10

c 3 1 2

j x x x

j x x x

j x x x

Do you notice any patterns

4

Letrsquos consider a function thatrsquos not easy to expand

sinj x x x

It would be nice to develop a method to differentiate the product of two functions

Explain how the function above can be modeled as j(x)=f(x)g(x)

5

The Derivative of a ProductWhen Hana looked at the function j(x)=f(x)g(x) she immediately thought of a rectangle whose length and width were f(x) and g(x) and whose area was j(x) Hana would like to find jrsquo(x) by using the definition of the derivative as a limit

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 3: The Product Rule

Try some more difficult cases

2 3

Find the derivative of

a 2 1 5

b 5 1 10

c 3 1 2

j x x x

j x x x

j x x x

Do you notice any patterns

4

Letrsquos consider a function thatrsquos not easy to expand

sinj x x x

It would be nice to develop a method to differentiate the product of two functions

Explain how the function above can be modeled as j(x)=f(x)g(x)

5

The Derivative of a ProductWhen Hana looked at the function j(x)=f(x)g(x) she immediately thought of a rectangle whose length and width were f(x) and g(x) and whose area was j(x) Hana would like to find jrsquo(x) by using the definition of the derivative as a limit

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 4: The Product Rule

4

Letrsquos consider a function thatrsquos not easy to expand

sinj x x x

It would be nice to develop a method to differentiate the product of two functions

Explain how the function above can be modeled as j(x)=f(x)g(x)

5

The Derivative of a ProductWhen Hana looked at the function j(x)=f(x)g(x) she immediately thought of a rectangle whose length and width were f(x) and g(x) and whose area was j(x) Hana would like to find jrsquo(x) by using the definition of the derivative as a limit

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 5: The Product Rule

5

The Derivative of a ProductWhen Hana looked at the function j(x)=f(x)g(x) she immediately thought of a rectangle whose length and width were f(x) and g(x) and whose area was j(x) Hana would like to find jrsquo(x) by using the definition of the derivative as a limit

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 6: The Product Rule

6

The Derivative of a Product

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 7: The Product Rule

7

The Derivative of a Product

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 8: The Product Rule

8

The Derivative of a Product

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 9: The Product Rule

9

The Derivative of a Product

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 10: The Product Rule

10

Using your result

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 11: The Product Rule

11

This is the Product Rule

It seems to work for simple binomials like 4 1 3 1

but what about other kinds of functions

f x x x

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 12: The Product Rule

12

Sneaky Product Rule Problems

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 13: The Product Rule

13

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment
Page 14: The Product Rule

14

Assignment

bullHW R

bull See you tmrrw

  • The Product Rule
  • Today you are going to find the derivative of functions of the
  • Try some more difficult cases
  • Letrsquos consider a function thatrsquos not easy to expand
  • The Derivative of a Product
  • The Derivative of a Product (2)
  • The Derivative of a Product (3)
  • The Derivative of a Product (4)
  • The Derivative of a Product (5)
  • Using your result
  • This is the Product Rule
  • Sneaky Product Rule Problems
  • Slide 13
  • Assignment

Product & Quotient Rule

Basic Counting. This Lecture We will study some basic rules for counting. Sum rule, product rule, generalized product rule Permutations, combinations

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Product and Quotient Rule - Illinois Institute of Technology · PDF fileThe Product and Quotient Rule . Agenda 1. The Product Rule Definition 2. The Product Rule Examples 3. The Quotient

The Building Block of Calculus - Chapter 4 The Product Rule and Quotient Rule

The Product Rule · The Product Rule We’re going to nd the derivative of products of functions, for example: The Product Rule We’re going to nd the derivative of products of functions,

Product Category Rule - NSF International · 2017-07-24 · Product Category Rule for Environmental Product Declarations Architectural Coatings: NAICS 325510 PRODUCT CATEGORY RULES

AP CALCULUS 1009 : Product and Quotient Rules. PRODUCT RULE FOR DERIVATIVES Product Rule: (In Words) ________________________________________________

MATH 136 More Derivatives: Product Rule, …MATH 136 More Derivatives: Product Rule, Quotient Rule, Chain Rule The Product Rule gives the formula for differentiating the product of

Product Category Rule Product Category Rule for ... · Product Category Rule Product Category Rule for Declaration for Power ... Functional Unit 1 square meter of door opening maintained

Product Rule HW3 - · PDF fileProduct Rule HW #3 Use the product rule to simplify each expression. 1) !!!∙! 2) 3!∙!! 3) 5!!∙2! 4) (7!! ... Microsoft Word - Product Rule HW3.docx

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The Product and Quotient Rules - University of Reading · Section 2: The Product Rule 5 2. The Product Rule The product rule states that if u and v are both functions of x and y is

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