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Product & Quotient Rules Higher Order Derivatives Lesson 3.3

Product & Quotient Rules Higher Order Derivatives

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Product & Quotient Rules Higher Order Derivatives. Lesson 3.3. Basic Rules. Product Rule. How would you put this rule into words?. Try Some More. Use additional rules to determine the derivatives of the following function. Basic Rule. Quotient Rule. How would you put this rule into words?. - PowerPoint PPT Presentation

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Page 1: Product & Quotient Rules Higher Order Derivatives

Product & Quotient RulesHigher Order Derivatives

Lesson 3.3

Page 2: Product & Quotient Rules Higher Order Derivatives

Basic Rules

• Product Rule

( ) ( ) ' ( ) '( ) '( ) ( )f x g x f x g x f x g x

2

( ) ( ) '( ) ( ) '( )'

( ) ( )

f x g x f x f x g x

g x g x

How would you put this rule into words?

How would you put this rule into words?

Page 3: Product & Quotient Rules Higher Order Derivatives

Try Some More

• Use additional rules to determine the derivatives of the following function

( ) xf x x e 3 2( ) 2 3 6 3p x x x

( ) 3 cos 4sinh x x x x

Page 4: Product & Quotient Rules Higher Order Derivatives

Basic Rule

• Quotient Rule

2

( ) ( ) '( ) ( ) '( )'

( ) ( )

f x g x f x f x g x

g x g x

How would you put this rule into words?

How would you put this rule into words?

Page 5: Product & Quotient Rules Higher Order Derivatives

A Memory Trick

• Given

• Then

2

( )( )

( )

( ) ( ) ( ) ( )'( )

( )x x

hi xf x

ho x

ho x D hi x hi x D ho xf x

ho x

Page 6: Product & Quotient Rules Higher Order Derivatives

Just Checking . . .

• Find the derivatives of the given functions

sin xy

x

4 2( ) 1

1f x x

x

2

7 4( )

5

xq x

x

Page 7: Product & Quotient Rules Higher Order Derivatives

Other Trig Derivatives

• Now try it out

2 2tan sec cot csc

sec sec tan csc csc cot

d dx x x x

dx dxd d

x x x x x xdx dx

4 tan ' ?f f

sec tand

x xdx

Page 8: Product & Quotient Rules Higher Order Derivatives

Higher-Order Derivatives

• Note that f ‘(x) is, itself a function– Possible to take the derivative of f ‘(x)

• This is called the second derivative

• Also possible to take higher derivatives

• Note TI capabilities

'( ) "( )d f x f x

Page 9: Product & Quotient Rules Higher Order Derivatives

Find Those High Orders

• Find the requested derivatives

2

2 32

4 1 ?d y

y x xdx

4 3 2( ) 2 9 6 5 '''( ) ?p x x x x p x

Page 10: Product & Quotient Rules Higher Order Derivatives

Assignment

• Lesson 3.3A• Page 147• Exercises 1 – 85 EOO

(Every Other Odd)

• Lesson 3.3B• Page 148 • Exercises 87 – 107 Odd