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. () ( ) ( ) ( ) 32 3 3 2 15 2 1 3 5 5 2 4 3 1 0 7 d xxx x x xx x dx ⎡⎤ ++ + = + + + + + = + + + ⎣⎦ 2. 33 2 3 2 1 21 2 1 3 2 3 1 22 2 dx x x x x dx ⎛⎞ + −= + +− = −+ + ⎜⎟ ⎢⎥ ⎝⎠ 3. ( )( ) ( )( ) ( )( ) 454 4 3 58 5 4 3 2 1 1 2 1 5 81 11 86581 d x x x x x x x x xxxx dx + + = + + = +−+− 4. ( ) ( ) 3 44 3 2 2 3 3 1 2 1 1 4 1 1 1 0 1 d x x x x dx x = + − + =+ + + + 5. ( ) ( ) ()( ) ( ) ( ) ( ) ( ) ( ) 43 4 3 4 3 2 2 2 2 2 2 14 1 2 1 1 8 1 1 1 9 1 d x x x x x x x x x dx + = + + += ++ += + + 6. 13 1 2 2 xx x x dx x = 7. ( )( ) ( ) ( ) ( ) ( ) ( ) 10 9 10 9 2 2 2 2 2 3 1 03 2 3 2 2 3 1 1 2 7 d x x x x x x x x dx +−= + + = + 8. ( ) ( ) ( ) () ( ) 3 3 2 3 3 2 3 2 26 3 4 3 3 42 63 2 6 636 1 1 2 63 6 1 8 63 4 23 dd dx dx x x x x x x x x x x =−+ + + + + 9. ( ) ( ) ( ) 2 1 1 3 51 1 2 5 1 5 2 3 x dx + + = + + 10. ( ) ( )( ) ( ) ( ) 74 7 4 3 64 41 0 7 6 31 2 1 2 2 1 1 2 7 2 1 3 12 1 45 108 7 d x x x x x x dx x x x = + + + 11. 1 ( 1)(1) ( 2 1 (1 ) ) x x dx x == + 12. ( ) () () ( ) 2 2 70 12 1 12 1 7 77 x dx + −− ⎣++ 13. ( ) ( ) 2 2 1 x dx x +
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Section 3.1 The Product and Quotient Rules 109 14. () () () 22 2 2 1 12 2 2 1 11 x xx x x dx x dx dx x x ⎡⎤ +− ⎢⎥ == = + ⎣⎦ ++ + 15. 224 4 3 2 3 (2 1) (1) ( 3)(2)(2 1)(2) 4 24 11 (2 1)(2 11) 2 11 (2 1) x x x x x x x x dx x x x + + + + + = = + + + 16. 43 3 4 3 3 4 3 4 4 3 44 4 4 4 4 d x x x x x dx x −− + =+ 17. ( ) () ( ) 2 2 10 22 4 0 1 dx x π += + = ⎣+ 18. ( )( ) ( )( ) cx d a ax b c d ax b ad bc dx cx d cx d cx d + + 19. ( ) ( ) 2 323 2 2 2 36 5 3 5 1 2 35 1 1 8 1 5 6561 02 3 33 52 0 1 5 5 ( 1 ) ( 3 ) x x x x x x x x dx x x x −+ + + +− − + + + 20. ( ) () ( ) ( ) 2 2 25 3 5 3 5 4 4 2 2 2 2 1 2 242 2 2 1 1 x x x x x x x x dx x x + = + + Alternatively, we can factor ( ) 2 21 + in the numerator to obtain ( ) ) ( ) ( ) ( ) ( ) 2 2 2 2 3 24 3 3 2 2 2 21 1 2 2 1 2 1 1 x x x x x x x x x + = + + 21. () ( )( ) 2 2 32 2 2 2 2 2 2 1 6 2 2 2 98 2 d x x x x dx x ⎛⎞ ++ − = + + − ⎜⎟ ⎝⎠ =− + + 22. 2 1 2 2 dd x dx dx x = = 23. 1 2 2 1 2 1 x dx dx x = + = + + 24. 1 2 13 23 3 2 3 3 2 3 31 1 313 1 3 1 1 x dx dx x = + = + +
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110 Chapter 3: Techniques of Differentiation 25. 42 32 22 43 3 3 343 4 dx x d x x xx x dx x dx x ⎡⎤ −+ −− =− + = = ⎢⎥ ⎣⎦ 26. () 12 2 11 1 1 1 1 dx x x x + −⋅ +⋅ + == = 27. ( ) 21 2 2 1 2 1 2 2 1 2(2 1) ( 2) (2 ( 2) (2)(2 1)(2) ( 2) 2 2 1 10 17 44 1 2 04 4 1 7 28 4 dd x x x x x x dx dx x x x ++ = + + = + + + + + =+ + + = = + 28. 1 (3 3 (1) 31 ( ) 23 2 1 x d x x dx x dx x x ⎛⎞ ⎜⎟ ⎝⎠ = 29. 52 ) (1 ) yx x + 4 4 ( 2) (2)( ( 1) (5)( 2) ( 2) ( 1)(7 dy xxx x x dx +++ − =− + + 3 88 x dy dx = = Equation of tangent line: y – 16 = 88( x – 3) or 88 248 30. 1 1 x y x + = ( 1)(1) ( 2 ) ) dy x x dx + 2 2 x dy dx = Equation of tangent line: y – 3 = –2( x – 2) or 27 + 31. 5 3 ) (4 ) x y x = 34 524 64 ) ( 5 ) )(2 ) ( 3 ) ) (2 ) ( 4 ) ) ) dy x x x x x x dx −−− The tangent line is horizontal when 0. dy dx = 4 4 4 ) ( 4 ) 0 ( 2) (2 14) 0 2 or 7 ) x x x =⇒ − − =⇒= =
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Section 3.1 The Product and Quotient Rules 111 32. 2 1 1 y x = + ( ) () () ( ) () 2 22 10 12 2 11 xx dy x dx +− == ++ ( ) ()( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 12 2 2 1 2 2 18 1 13 0 when 2 1 8 1 0 2 1 8 1 3 3 x x x x x dy dx x x x dx + + + + ⎡⎤ ⎢⎥ ⎣⎦ =− + = + =+ = ± = ± Inflection points are at 33 , 34 and , .
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This note was uploaded on 04/24/2010 for the course MATH 9374 taught by Professor Smith during the Spring '10 term at University of California, Berkeley.

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

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