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11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
11 x1 t16 06 derivative times function (2013)
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11 x1 t16 06 derivative times function (2013)

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  • 1.  f  x  f  x  dx n
  • 2.  f  x  f  x  dx n  f  x n1  c  f  x  f  x  dx  n n 1
  • 3.  f  x  f  x  dx n  f  x n1  c  f  x  f  x  dx  n n 1 d e.g. i  a) Find dx  1 x  3
  • 4.  f  x  f  x  dx n  f  x n1  c  f  x  f  x  dx  n n 1 d e.g. i  a) Find dx d dx  1 x  3   1  1 1  x 3  1  x 3  2  3 x 2  2  3x 2  2 1  x3
  • 5. b) Hence find;  x2 dx 3 1 x
  • 6. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx dx    3 3 3 2 1 x 1 x
  • 7. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3
  • 8. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx
  • 9. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx 1   2 x 2  x 2 dx 2
  • 10. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3
  • 11. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3
  • 12. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx 
  • 13. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u  2  x2
  • 14. b) Hence find;   x2 dx 3 1 x x2 2  3x 2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u  2  x2 du  2x dx du  2 xdx
  • 15. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx dx    3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u u  2  x2 du  2x dx du  2 xdx
  • 16. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx    dx 3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  1 du 2 u u  2  x2 du  2x dx du  2 xdx
  • 17. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx    dx 3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u 1 du 2 1 1 2   u du 2 u  2  x2 du  2x dx du  2 xdx
  • 18. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx    dx 3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u 1 du 2 1 1 2   u du 2 3 1 2 2   u c 2 3 u  2  x2 du  2x dx du  2 xdx
  • 19. b) Hence find;   x2 dx 3 1 x 2  3x 2 x2 dx    dx 3 3 3 2 1 x 1 x 2   1  x3  c 3 ii   x 2  x 2 dx OR 1   2 x 2  x 2 dx 2 3 1 2   2  x 2 2  c 2 3 1  2  x 2  2  x 2  c 3 x 2  x 2 dx  u  2  x2 du  2x dx du  2 xdx u 1 du 2 1 1 2   u du 2 3 1 2 2   u c 2 3 1  2  x 2  2  x 2  c 3
  • 20. 1 iii  0 x x2 3  2 3 dx
  • 21. 1 iii  0 x 1 x2 3  2 3 dx 1 3x 2   dx 3 3 0 x 3  2 
  • 22. 1 iii  0 x x2 3  2 3 dx 1 1 3x 2   dx 3 3 0 x 3  2  1 3 1 2 3   3 x x  2  dx 30
  • 23. 1 iii  0 x x2 3  2 3 dx 1 1 3x 2   dx 3 3 0 x 3  2  1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6  
  • 24. 1 iii  0 x x2 3  2 3 dx 1 1 3x 2   dx 3 3 0 x 3  2  1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6   1 1 1      2 6  1  2 2  1  8
  • 25. 1 iii  0 x 1 x2 3  2 3 dx 1 1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6  1 1 1      2 6  1  2 2  1  8  x 0 1 3x 2   dx 3 3 0 x 3  2   OR x2 3  2 3 dx
  • 26. 1 iii  0 x 1 x2 3  2 3 dx 1 1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6  1 1 1      2 6  1  2 2  1  8  x 0 1 3x 2   dx 3 3 0 x 3  2   OR x2 3  2 3 dx u  x3  2 du  3 x 2 dx
  • 27. 1 iii  0 x 1 x2 3  2 3 dx 1 1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6  1 1 1      2 6  1  2 2  1  8  x 0 1 3x 2   dx 3 3 0 x 3  2   OR x2 3  2 3 dx u  x3  2 du  3 x 2 dx x  0, u  2 x  1, u  1
  • 28. 1 iii  0 x 1 x2 3  2 3 dx 1 1 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6  1 1 1      2 6  1  2 2  1  8  x 0 1 3x 2   dx 3 3 0 x 3  2   OR 1 x2 3  2 3 1 3   u du 3 2 dx u  x3  2 du  3 x 2 dx x  0, u  2 x  1, u  1
  • 29. 1 iii  0 x 1 x2 3  2 3 dx OR  x 0 x2 3  2 3 1 1 1 3   u du 3 2 1 1  2 1   u  2 6 1 3x 2   dx 3 3 0 x 3  2  3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6   1 1 1      2 6  1  2 2  1  8 dx u  x3  2 du  3 x 2 dx x  0, u  2 x  1, u  1
  • 30. 1 iii  0 x 1 x2 3  2 3 dx OR  x 0 x2 3  2 3 dx 1 1 1 3   u du 3 2 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6 1 1  2 1   u  2 6 1 1 1      2 6  1  2 2  1 1 1      2 6  1  2 2  1  8 1 3x 2   dx 3 3 0 x 3  2   1  8  u  x3  2 du  3 x 2 dx x  0, u  2 x  1, u  1
  • 31. 1 iii  0 x 1 x2 3  2 3 dx OR  x 0 x2 3  2 3 dx 1 1 1 3   u du 3 2 3 1 2 3   3 x x  2  dx 30 2 1 1 3   x  2  0 6 1 du  3 x 2 dx x  0, u  2 x  1, u  1 1  2 1   u  2 6 1 1 1      2 6  1  2 2  1 1 1      2 6  1  2 2  u  x3  2 1  8 1 3x 2   dx 3 3 0 x 3  2    1  8 Exercise 11H; 1, 3, 5, 7ace etc, 8bdf,9 11*

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