chapter 3 - Chapter 3 Exercises 3.1 1 d x 1 x 3 5 x 2 dx =...

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131 Chapter 3 Exercises 3.1 1. d dx [( x + 1)( x 3 + 5 x + 2)] = ( x + 1)(3 x 2 + 5) + ( x 3 + 5 x + 2) = 4 x 3 + 3 x 2 + 10 x + 7 2. 35 (3 1 1 ) (1 ) d xx x dx ⎡⎤ ++ ⎣⎦ 34 2 43 2 ( 3 11)(5)( 1) (3 3)( ( 1) (8 3 18 52) xx x x x x x =++ −+ + − =− −++ 5 3. 45 44 3 8543 (2 1)( 1)( 5 ) (8 1)( 1 86581 d x dx x x x xxxx −+ − + + −+− + + + 5 4. 2 2 2 ) ( 2 1 ) ( 3)(2) (2 )(2 626 d dx +− =+ + =−+ x 5. 24 ) d dx + 23 2 22 3 2 4 232 (4)( 1) (2 ) (1)( 8( 1 ) ( 1 ) ) ( 91 ) x x x + =+ + + + 4 6. (7 d dx (2)(7 1)(7) (2 )(7 2( 7 1 ) ( 1 4 1 ) x x x + =−− 7. d dx [( x 2 + 3)( x 2 3) 10 ] = ( x 2 + 3)(10)( x 2 3) 9 x ) + ( x 2 3) 10 x ) = 2 x ( x 2 3) 9 (11 x 2 + 27) 8. 4 [( 1)(7 2)] d dx −+ [ ] 3 4[( 1)(7 2)] ( 1)(7) (1)(7 2) x x =− + + [] 3 (56 20) ( 1)(7 2) x
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Chapter 3: Techniques of Differentiation ISM: Calculus & Its Applications, 11e 132 9. d dx (5 x + 1)( x 2 1) + 2 x + 1 3 = x + 1)(2 x ) + (5)( x 2 1) + 1 3 (2) = 15 x 2 + 2 x 13 3 10. 74 2 (3 12 d xx x dx ⎡⎤ +− ⎣⎦ 3 6 4 (2)(3 12 1)(12 12) 7 (3 12 xxx x x =+ + + + 2 6 41 0 7 (3 12 1)(45 108 7 ) x xxx x =+− + − 11. d dx x 1 x + 1 = ( x + 1)(1) ( x ( x + 2 = 2 ( x + 2 12. d dx x 2 + 2 x 1 x 2 + 2 x 2 = ( x 2 + 2 x 2)(2 x + 2) ( x 2 + 2 x x + 2) ( x 2 + 2 x 2) 2 = 2 x 2 ( x 2 + 2 x 2) 2 13. d dx x 2 1 x 2 + 1 = ( x 2 + 1)(2 x ) ( x 2 x ) ( x 2 + 2 = 4 x ( x 2 + 2 14. d dx 1 x + 1 + 1 x 1 = d dx 2 x x 2 1 = ( x 2 1)(2) (2 x )(2 x ) ( x 2 1) 2 = 2 x 2 2 ( x 2 2 15. 2 22 2 3 ( 21 ) ( 1 ) (3 ) ( ) ( 2 ) ( dx x x x dx ++ + + = ⎢⎥ 2 2 42 4 1 1) x =− + 1 16. 2 2 2 11 2 (2 1)(2) (2 11)(2) 72 1 7 ( 2 1 ) x x dx x x −+ = 2 48 7(2 x = +
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ISM: Calculus & Its Applications, 11e Chapter 3: Techniques of Differentiation 133 17. d dx 1 π + 2 x 2 + 1 = 0 + ( x 2 + 1)(0) 2(2 x ) ( x 2 + 1) 2 = 4 x ( x 2 + 1) 2 18. d dx 7 9 + x 2 + x + 1 x 5 + 1 = 0 + ( x 5 + 1)(2 x + 1) ( x 2 + x + 1)(5 x 4 ) ( x 5 + 2 = 3 x 6 4 x 5 5 x 4 + 2 x + 1 ( x 5 + 2 19. d dx 3 x 2 + 5 x + 1 3 x 2 = (3 x 2 )(6 x + 5) x 2 + 5 x + 1)( 2 x ) x 2 ) 2 = 5( x + 1)( x + 3) x 2 ) 2 20. 2 22 (1 ) dx dx x ⎡⎤ ⎢⎥ + ⎣⎦ 2 2 2 ( 1) (2 ) (2)( 1)(2 ) ) x xx x x x +− + = + 5 24 ) x x x = + 21. ( ) 2 2 (3 2 2)( 2) 2 (3 2 2)( 2) (3 2 2)(1) (6 2)( 2) 2( 2)(3 2 2)(9 8 2) d xx x dx x x x ++ =+ + + + + + =− −− 22. 1 2 3 2 3 2 1 (2 ) 2 1 ) ( 1 ) 2 1 2( 2) dd x dx dx x x x =
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Chapter 3: Techniques of Differentiation ISM: Calculus & Its Applications, 11e 134 23. 1 1 2 1 1 1 dd x dx dx x ⎛⎞ ⎡⎤ =+ ⎜⎟ ⎢⎥ + ⎣⎦⎝⎠ () 1 2 2 1 1 2 xx =− + ⎝⎠ 2 1 21 + 24. 1 1 3 3 3 31 1 x dx dx x + ⎣⎦ ⎝⎠ 2 2 3 3 1 3 + 2 2 3 3 1 1 + 25. d dx x 4 4 x 2 + 3 x = d dx x 3 4 x + 3 x = 3 x 2 4 3 x 2 26. 2 (3 ) (4 ) dx dx x x + ++ ⎣⎦ [ ] [] 2 ( 3)( 4)(1) ( 2) ( 3)(1) (1)( 4) ) ) x x x ++ − + ++ + = 2 22 42 ( 7 12) 27. 1 2 2(2 1) ( 2) (2 1) x x dx dx ++= + + 11 2 1 ( 2) (2)(2 1)(2) ( 2) (2 1) 2 xx x x + + + + 2 20 44 17 x = + 28. 1 2 ( ) dx x dx x −− = 2 1 (3 (3 1) (1) 2 x x = 2 25 231 x =
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ISM: Calculus & Its Applications, 11e Chapter 3: Techniques of Differentiation 135 29. y = ( x 2) 5 ( x + 1) 2 dy dx = ( x 2) 5 (2)( x + + ( x + 1) 2 (5)( x 2) 4 = ( x 2) 4 ( x + 1)(7 x + dy dx x = 3 = 88 Equation of tangent line: y – 16 = 88( x – 3); 88 248 yx = 30. y = x + 1 x 1 dy dx = ( x 1)(1) ( x + ( x 1) 2 = 2 ( x 2 dy dx x = 2 =− 2
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This note was uploaded on 04/04/2010 for the course MATH MATH 16B taught by Professor Unknown during the Spring '10 term at UC Davis.

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chapter 3 - Chapter 3 Exercises 3.1 1 d x 1 x 3 5 x 2 dx =...

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