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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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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