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04_M_GOLD_9004_12_ch03-1

# 04_M_GOLD_9004_12_ch03-1 - Chapter 3 Techniques of...

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108 Chapter 3 Techniques of Differentiation 3.1 The Product and Quotient Rules 1. ( ) ( ) ( ) ( ) ( ) 3 2 3 3 2 1 5 2 1 3 5 5 2 4 3 10 7 d x x x x x x x x x x dx + + + = + + + + + = + + + 2. ( ) ( ) ( ) 3 3 2 3 2 1 2 1 2 1 3 2 3 1 2 2 2 d x x x x x x x dx + = − + + = − + + 3. ( )( ) ( )( ) ( )( ) 4 5 4 4 3 5 8 5 4 3 2 1 1 2 1 5 8 1 1 18 6 5 8 1 d x x x x x x x x x x x x dx + + = + + + = − + + 4. ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 4 4 3 2 2 2 2 3 2 2 1 1 3 1 2 1 1 4 1 1 1 1 10 1 d x x x x x x x x x x dx x x x x x + + = + + + + + + = + + + + 5. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 3 4 3 4 3 2 2 2 2 2 2 2 2 1 4 1 2 1 1 8 1 1 1 9 1 d x x x x x x x x x x x dx + = + + + = + + + = + + 6. ( ) 1 3 1 2 2 d x x x x x dx x = + = 7. ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 10 9 10 9 2 2 2 2 2 2 2 3 3 3 10 3 2 3 2 2 3 11 27 d x x x x x x x x x x dx + = + + = + 8. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 3 3 3 3 3 3 3 2 3 3 3 3 2 3 3 3 2 2 6 3 4 2 6 3 2 6 3 4 2 6 3 2 6 6 3 6 1 4 2 6 3 12 6 36 18 6 3 4 2 6 3 48 18 12 3 d d x x x x x x x x x dx dx x x x x x x x x x x x x x x x x x x x x x + = + + = + + + + = + + + + = + + + 9. ( ) ( ) ( )( ) ( ) ( ) 2 2 2 2 1 1 13 5 1 1 5 1 2 5 1 2 15 2 3 3 3 d x x x x x x x x dx + + + = + + + = + 10. ( ) ( ) ( )( ) ( ) ( )( ) 2 2 7 4 7 4 3 6 4 4 10 7 6 3 12 1 2 3 12 1 12 12 7 3 12 1 3 12 1 45 108 7 d x x x x x x x x x x dx x x x x x + = + + + + = + + 11. 2 2 1 ( 1)(1) ( 1)(1) 2 1 ( 1) ( 1) d x x x dx x x x + = = + + + 12. ( ) ( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 7 0 1 2 1 1 2 1 7 7 7 x x x d x dx x x x x x x + + + = = + + + + + + 13. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 1 2 1 4 1 1 1 x x x x d x x dx x x x + = = + + +

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Section 3.1 The Product and Quotient Rules 109 14. ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 x x x x x d x d x x dx dx x x x x + = = = + + + + 15. 2 2 2 2 4 4 3 2 3 (2 1) (1) ( 3)(2)(2 1)(2) 4 24 11 (2 1)(2 11) 2 11 (2 1) (2 1) (2 1) (2 1) (2 1) d x x x x x x x x x dx x x x x x + + + + + + + + + = = − = − = − + + + + + 16. 4 3 3 4 3 3 4 3 4 4 3 1 1 4 4 4 4 4 4 d x x x x x x x dx x + = + = + = + 17. ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 0 2 2 1 2 4 0 1 1 1 x x d x dx x x x π + + = + = + ⎦ + + 18. ( )( ) ( )( ) ( ) ( ) 2 2 cx d a ax b c d ax b ad bc dx cx d cx d cx d + + + = = + + + 19. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 3 6 5 3 5 1 2 3 5 1 18 15 6 5 6 10 2 3 3 3 5 20 15 5( 1)( 3) 3 3 x x x x x d x x x x x x x x dx x x x x x x x x x + + + + + + + + + = = + + + + = = 20. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 5 3 5 3 5 2 2 4 4 2 2 2 2 2 1 2 2 1 2 2 4 2 4 4 2 2 1 1 1 1 x x x x x d x x x x x x x x dx x x x x + + + + = = = + + + + Alternatively, we can factor ( ) 2 2 1 x x + in the numerator to obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 3 2 4 3 3 2 2 2 2 2 2 1 1 2 1 2 2 1 2 2 1 2 2 1 1 1 1 x x x x x x x x x x x x x x x x x + + +
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04_M_GOLD_9004_12_ch03-1 - Chapter 3 Techniques of...

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