11X1 T08 03 rules for differentiation

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<ul><li> 1. Rules For Differentiation </li></ul><p> 2. Rules For Differentiation (1) y c 3. Rules For Differentiation (1) y c f x c 4. Rules For Differentiation (1) y c f x cf x h c 5. Rules For Differentiation (1) y ccc f x cf x lim h0 hf x h c 6. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h 7. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 8. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kx 9. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kx f x kx 10. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kx f x kx f x h k x h kx kh 11. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kxkx kh kx f x kxf x limh0h f x h k x h kx kh 12. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kxkx kh kx f x kxf x limh0h f x h k x hkh lim kx kh h0 h 13. Rules For Differentiation (1) y c cc f x cf x lim h0 hf x h c0 lim h 0 h lim 0h00 (2) y kxkx kh kx f x kxf x limh0h f x h k x hkh lim kx kh h0 h lim kh0k 14. (3) y x n 15. (3) y x na 2 b 2 a b a b 16. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 17. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 18. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 19. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xn 20. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xn f x h x hn 21. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xnf x h x hn x h xn nf x lim h0 h 22. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xnf x h x hn x h xn nf x lim h0 h lim x h x x h x h x x h x n2 x n1 n 1 n2 h0 h 23. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xnf x h x hn x h xn nf x lim h0 h lim x h x x h x h x x h x n2 x n1 n 1 n2 h0 h lim h x h x h x x h x n2 x n1n 1n2h0 h 24. (3) y x na 2 b 2 a b a b a 3 b3 a b a 2 ab b 2 a 4 b 4 a b a 3 a 2b ab 2 b3 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 f x xnf x h x hn x h xn nf x lim h0 h lim x h x x h x h x x h x n2 x n1 n 1 n2 h0 h lim h x h x h x x h x n2 x n1n 1n2h0 h lim x h x h x x h x n2 x n1 n 1n2 h0 25. f x lim x h x h x x h x n2 x n1 n 1n2h0 26. f x lim x h x h x x h x n2 x n1 n 1n2h0 x n1 x n1 x n1 x n1 27. f x lim x h x h x x h x n2 x n1 n 1n2h0 x n1 x n1 x n1 x n1 nx n1 28. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x 29. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x1 f x x 30. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x1 f x x1f x h xh 31. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x 1 11 f x f x lim x h xxh0 h1f x h xh 32. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x 111 f x f x lim x h xxh0h1 xxhf x h lim xh h0 hx x h 33. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x 111 f x f x lim x h xx h0 h1 xxhf x h lim xh h0 hx x h h lim h0 hx x h 34. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 f x f x lim x h xx h0h1 xxhf x h lim xh h0 hx x h h lim h0 hx x h 1 lim h0 x x h 35. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 f x f x lim x h xx h0h1 xxhf x h lim xh h0 hx x h h lim h0 hx x h 1 lim h0 x x h 1 2 x 36. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 Note: f x f x lim x h xx h0h1 xxhf x h lim xh h0 hx x h h lim h0 hx x h 1 lim h0 x x h 1 2 x 37. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 Note: f x f x lim x h xx h0h f x x 11 xxhf x h lim xh h0 hx x h h lim h0 hx x h 1 lim h0 x x h 1 2 x 38. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 Note: f x f x lim x h xx h0h f x x 11 xxhf x h lim xh h0 hx x h f x x 2h lim h0 hx x h 1 lim h0 x x h 1 2 x 39. f x lim x h x h x x h x n2 x n1n 1n2 h0 x n1 x n1 x n1 x n1 nx n1 1 (4) y x111 Note: f x f x lim x h xx h0h f x x 11 xxhf x h lim xh h0 hx x h f x x 2h1 lim 2 h0 hx x h x 1 lim h0 x x h 1 2 x 40. (5) y x 41. (5) y x f x x 42. (5) y x f x x f x h x h 43. (5) y x f x xxh xf x h x h f x lim h 0 h 44. (5) y x f x xxh x xh xf x h x h f x lim h 0 hxh x 45. (5) y x f x xxh x xh xf x h x h f x lim h 0hxh x xhx limh0 h xh x 46. (5) y x f x xxh x xh xf x h x h f x lim h 0hxh x xhx limh0 h xh xh limh0 h xh x 47. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0 h xh x h lim h0 h xh x1 lim h0 xh x 48. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0 h xh x h lim h0 h xh x1 lim h0 xh x1 x x 49. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0 h xh x h lim h0 h xh x1 lim h0 xh x 1 x x 12 x 50. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0 h xh x h lim h0 h xh x Note: 1 lim h0 xh x 1 x x 12 x 51. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0h xh x h lim h0h xh xNote:11 f x x2 limh0 xh x 1 x x 12 x 52. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0h xh x h lim h0h xh x Note: 11f x x2 limh0 xh x 1 1f x x 21 2x x1 2 x 53. (5) y x f x xxh xxh xf x h x h f x limh 0 hxh xxhx lim h0h xh x h lim h0h xh x Note: 11f x x2 limh0 xh x 1 1f x x 21 2 1x x2 x1 2 x 54. e.g. i y 7 55. e.g. i y 7 dy0 dx 56. e.g. i y 7 dy0 dx ii y 37 x 57. e.g. i y 7 dy0 dx ii y 37 xdy 37dx 58. e.g. i y 7 dy0 dx ii y 37 xdy 37dx iii y x10 59. e.g. i y 7 dy0 dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx 60. e.g. i y 7 dy0 dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2 61. e.g. i y 7 dy0 dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 62. e.g. i y 7 v y 2 x 12dy0 dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 63. e.g. i y 7 v y 2 x 12dy0 4x2 4 x 1 dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 64. e.g. i y 7 v y 2 x 12dy0 4x2 4 x 1 dx dy 8x 4dx ii y 37 xdy 37dx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 65. e.g. i y 7 v y 2 x 12dy0 4x2 4 x 1 dxdy 8x 4 dx ii y 37 x 1dy vi y 3x 2 37xdx iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 66. e.g. i y 7 v y 2 x 1 2dy0 4x2 4 x 1 dxdy 8x 4 dx ii y 37 x1dy vi y 3x 2 37 xdx 3 x x 2 iii y x10dy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 67. e.g. i y 7 v y 2 x 1 2dy0 4x2 4 x 1 dxdy 8x 4 dx ii y 37 x1dy vi y 3x 2 37 xdx 3 x x 2 dy 3 2 x 3 iii y x10 dxdy 10 x 9dx iv y 3x 2 6 x 2dy 6x 6dx 68. e.g. i y 7 v y 2 x 1 2dy0 4x2 4 x 1 dxdy 8x 4 dx ii y 37 x1dy vi y 3x 2 37 xdx 3 x x 2 dy 3 2 x 3 iii y x10 dxdy 2 10 x 9 3 3dx x iv y 3x 2 6 x 2dy 6x 6dx 69. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 1 dxdy 8x 4 dx ii y 37 x1dy vi y 3x 2 37 xdx 3 x x 2 dy 3 2 x 3 iii y x10 dxdy 2 10 x 9 3 3dx x iv y 3x 2 6 x 2dy 6x 6dx 70. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dxx 2 dy 8x 4 dx ii y 37 x1dy vi y 3x 2 37 xdx 3 x x 2 dy 3 2 x 3 iii y x10 dxdy 2 10 x 9 3 3dx x iv y 3x 2 6 x 2dy 6x 6dx 71. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 1 5dx x 2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 37 xdx 3 x x 2 dy 3 2 x 3 iii y x10 dxdy 2 10 x 9 3 3dx x iv y 3x 2 6 x 2dy 6x 6dx 72. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 1 5dx x 2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dxdy 2 10 x 9 3 3dx x iv y 3x 2 6 x 2dy 6x 6dx 73. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dx x2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dx viii If f x x3 3, dy 10 x 9 3 3 2find f 2 dxx iv y 3x 2 6 x 2dy 6x 6dx 74. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dx x2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dx viii If f x x3 3, dy 10 x 9 3 3 2find f 2 dxxf x x3 3 iv y 3x 2 6 x 2dy 6x 6dx 75. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dx x2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dx viii If f x x3 3, dy 10 x 9 3 3 2find f 2 dxxf x x3 3 iv y 3x 2 6 x 2 f x 3x 2dy 6x 6dx 76. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dx x2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dx viii If f x x3 3, dy 10 x 9 3 3 2find f 2 dxxf x x3 3 iv y 3x 2 6 x 2 f x 3x 2dy 6x 6 f 2 3 22dx 77. e.g. i y 7 v y 2 x 1 2 vii y x 2 x dy0 4x2 4 x 15dx x2 dy 8x 4dy 5 3 dx x2 ii y 37 x1 dx 2dy vi y 3x 2 5 37 x x xdx 3 x x 2 2 dy 3 2 x 3 iii y x10 dx viii If f x x3 3, dy 10 x 9 3 3 2find f 2 dxxf x x3 3 iv y 3x 2 6 x 2 f x 3x 2dy 6x 6 f 2 3 22dx 12 78. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 79. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1y 5 x3 6 x 2 2 80. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1y 5 x3 6 x 2 2 dy 15 x 2 12 x dx 81. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1y 5 x3 6 x 2 2 dy 15 x 2 12 x dx dywhen x 1, 15 1 12 12dx 82. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1y 5 x3 6 x 2 2 dy 15 x 2 12 x dx dywhen x 1, 15 1 12 12dx 3 83. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1y 5 x3 6 x 2 2 dy 15 x 2 12 x dx dywhen x 1, 15 1 12 12dx 3 required slope 3 84. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1 dy 15 x 2 12 x dx dywhen x 1, 15 1 12 12dx 3 required slope 3 85. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1 dyy 1 3x 3 15 x 2 12 x dx dywhen x 1, 15 1 12 12dx 3 required slope 3 86. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1 dyy 1 3x 3 15 x 2 12 x dx dywhen x 1, 15 1 12 12 3x y 2 0 dx 3 required slope 3 87. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal 88. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal y x 3 12 x 89. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal y x 3 12 xdy 3 x 2 12dx 90. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal dytangents are horizontal when0 y x 3 12 xdxdy 3 x 2 12dx 91. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal dytangents are horizontal when0 y x 3 12 xdx i.e. 3 x 12 0 2 dy 3 x 2 12dx 92. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12 x where the tangentsare horizontal dytangents are horizontal when0 y x 3 12 xdx i.e. 3 x 12 0 2 dy 3 x 2 12 x2 4dx x 2 93. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2 at the point 1,1 y 5 x3 6 x 2 2 y 1 3 x 1dyy 1 3x 3 15 x 2 12 xdx dywhen x 1, 15 1 12 1 23x y 2 0 dx3 required slope 3 x Find the points on the curve y x3 12...</p>