Product & Quotient RulesHigher Order Derivatives

Lesson 3.3

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?

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

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?

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

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

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

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

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

Assignment

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

(Every Other Odd)

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