Chapter11.3 - 2 2 ) 1 2 ( 2 2 4 ) 1 2 ( ) 2 )( ( ) 1 2 )( 2...

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Chapter 11.3 Product and Quotient Page 603 Theorem 1: ( 29 ) ( ' ) ( ) ( ) ( ' ' ) ( ) ( x g x f x g x f x g x f × + × = × Example 1. Find f’ ( x ) for ) 2 3 ( 2 ) ( 4 2 - = x x x f ) 12 )( 2 ( ) 2 3 )( 4 ( )' 2 3 )( 2 ( ) 2 3 ( )' 2 ( ) ( ' 3 2 4 4 2 4 2 x x x x x x x x x f + - = - + - = x x x x x 8 36 24 8 12 5 5 5 - = + - = Page 604 Example 2. Given ) 6 )( 9 2 ( ) ( 2 + - = x x x f (A) Find an equation of the tangent line at x = 3. ) 2 )( 9 2 ( ) 6 )( 2 ( )' 6 )( 9 2 ( ) 6 ( )' 9 2 ( ) ( ' 2 2 2 x x x x x x x x f - + + = + - + + - = 12 18 6 18 4 12 2 2 2 2 + - = - + + = x x x x x At x = 3: the slope : 12 12 54 54 12 ) 3 ( 18 ) 3 ( 6 ) 3 ( ' 2 = + - = + - = = f m the point : 45 ) 15 )( 3 ( ) 6 ) 3 )(( 9 ) 3 ( 2 ( ) 3 ( 2 - = - = + - = f , (3, – 45) the point-slope form of the tangent line: ) 3 ( 12 45 - = + x y . (B) Find the value(s) of x where the tangent line is horizontal, i.e. the slope m = 0. Solve for f’ ( x ) = 0: 2 , 1 0 ) 2 )( 1 ( 6 0 ) 2 3 ( 6 0 12 18 6 2 2 = = = - - = + - = + - x x x x x x x x g g g The solution(s) to f’ ( x ) = 0 is/are critical value(s) of f ( x ). 1
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Page 605 Theorem 2: [ ] 2 ' ) ( ) ( ' ) ( ) ( ) ( ' ) ( ) ( x g x g x f x g x f x g x f × - × = . Page 606 Example 4. Find the derivatives of the following functions: (A) 1 2 ) ( 2 - = x x x f 2 2 2 2 2 2 2 2 2 2 ) 1 2 (
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Unformatted text preview: 2 2 ) 1 2 ( 2 2 4 ) 1 2 ( ) 2 )( ( ) 1 2 )( 2 ( ) 1 2 ( )' 1 2 )( ( ) 1 2 ( )' ( ) ( '--=---=---=----= x x x x x x x x x x x x x x x x x f (B) 1 3 2 +-= t t t y 2 3 3 4 2 3 2 2 3 2 3 3 2 3 2 ) 1 ( 1 2 2 ) 1 ( ) 3 )( ( ) 1 )( 1 2 ( ) 1 ( )' 1 )( ( ) 1 ( )' ( ' +-+ +-= +--+-= + +--+-= t t t t t t t t t t t t t t t t t y (C) 3 4 4 3 3 4 2 2 2 2 2 2 2 2 2 2 6 6 ) 6 2 ( 2 ) 2 )( 3 ( ) )( 2 ( ) ( )' )( 3 ( ) ( )' 3 ( 3 x x x x x x x x x x x x x x x x x x x dx d = =--=--=---= -2...
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Chapter11.3 - 2 2 ) 1 2 ( 2 2 4 ) 1 2 ( ) 2 )( ( ) 1 2 )( 2...

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