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12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
12 x1 t02 01 differentiating exponentials (2014)
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12 x1 t02 01 differentiating exponentials (2014)

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  • 1. Exponentials
  • 2. Exponentials y x
  • 3. Exponentials y y  ax a  1 1 x
  • 4. Exponentials ya 0  a  1 x y y  ax a  1 1 x
  • 5. Exponentials ya 0  a  1 x y 1 y  ax a  1  y  ax   0  a  1    x
  • 6. Exponentials ya 0  a  1 x y x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x
  • 7. Exponentials ya 0  a  1 x y x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x domain : all real x
  • 8. Exponentials ya 0  a  1 x y x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x domain : all real x range : y  0
  • 9. Exponentials ya 0  a  1 x y x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x domain : all real x range : y  0 y  ex
  • 10. ya x Exponentials y 0  a  1 x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x domain : all real x range : y  0 y  ex   1 e is an irrational number, it is defined as; lim1  n   n n
  • 11. ya x Exponentials y 0  a  1 x y  a    a  1  1 y  ax a  1  y  ax   0  a  1    x domain : all real x range : y  0 y  ex   1 e is an irrational number, it is defined as; lim1  n   n n e  2.718281828
  • 12. ya x Exponentials y 0  a  1 x y  a    a  1  1 y  ax a  1   y  ax  0  a  1    x domain : all real x range : y  0 y  ex 1  1  e is an irrational number, it is defined as; lim  n   n n e  2.718281828 e.g. e3  20.086 (to 3 dp)
  • 13. Differentiating Exponentials
  • 14. Differentiating Exponentials y  e f x
  • 15. Differentiating Exponentials y  e f x dy  f  x e f  x  dx
  • 16. Differentiating Exponentials y  e f x dy  f  x e f  x  dx y  a f x
  • 17. Differentiating Exponentials y  e f x dy  f  x e f  x  dx y  a f x dy  f  x log a a f  x  dx
  • 18. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x y  a f x dy  f  x log a a f  x  dx
  • 19. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x dy  ex dx y  a f x dy  f  x log a a f  x  dx
  • 20. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x dy  ex dx ii  y  e5 x y  a f x dy  f  x log a a f  x  dx
  • 21. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x dy  ex dx ii  y  e5 x dy  5e5 x dx y  a f x dy  f  x log a a f  x  dx
  • 22. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x dy  ex dx ii  y  e5 x dy  5e5 x dx y  a f x dy  f  x log a a f  x  dx iii  y  e 4 x 3
  • 23. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e x dy  ex dx ii  y  e5 x dy  5e5 x dx y  a f x dy  f  x log a a f  x  dx iii  y  e 4 x 3 dy  4e 4 x  3 dx
  • 24. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e dy  ex dx x ii  y  e 5x dy  5e5 x dx y  a f x dy  f  x log a a f  x  dx iii  y  e 4 x 3 dy  4e 4 x  3 dx iv  y  e x 2 3 x  2
  • 25. Differentiating Exponentials y  e f x dy  f  x e f  x  dx e.g. i  y  e dy  ex dx x ii  y  e 5x dy  5e5 x dx y  a f x dy  f  x log a a f  x  dx iii  y  e 4 x 3 dy  4e 4 x  3 dx iv  y  e x 2 3 x  2 dy x 2 3 x  2  2 x  3e dx
  • 26. (v) y  3 x 2 e 4 x
  • 27. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx
  • 28. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1
  • 29. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 vi  y  e  2 3x 7
  • 30. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx
  • 31. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2 
  • 32. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2 
  • 33. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2 
  • 34. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2 
  • 35. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2  viii  Find the tangent to y  e 2 x  1 at the point 1, e 2  1
  • 36. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2  viii  Find the tangent to y  e 2 x  1 at the point 1, e 2  1 y  e2 x  1 dy  2e 2 x dx
  • 37. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2  viii  Find the tangent to y  e 2 x  1 at the point 1, e 2  1 y  e2 x  1 dy  2e 2 x dx dy when x  1,  2e 2 dx
  • 38. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2  viii  Find the tangent to y  e 2 x  1 at the point 1, e 2  1 y  e2 x  1 dy  2e 2 x dx dy when x  1,  2e 2 dx y  e 2  1  2e 2  x  1
  • 39. (v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx  12 x 2 e 4 x  6 xe 4 x  6 xe 4 x 2 x  1 ex vii  y  x e 3 dy e x  3e x   e x e x   2 x dx e  3 e 2 x  3e x  e 2 x  2 x e  3  e 3e x x  3 2 vi  y  e  2 3x 7 6 dy 3x  7e  2  3e3 x  dx 6 3x 3x  21e e  2  viii  Find the tangent to y  e 2 x  1 at the point 1, e 2  1 y  e2 x  1 dy  2e 2 x dx dy when x  1,  2e 2 dx y  e 2  1  2e 2  x  1 y  e 2  1  2e 2 x  2e 2 2e 2 x  y  e 2  1  0
  • 40. ix  y  4 x2
  • 41. ix  y  4 x2 dy x2  2 xlog 4 4 dx
  • 42. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x
  • 43. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey
  • 44. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey dx  ey dy
  • 45. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey dx  ey dy dy 1  y dx e
  • 46. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey dx  ey dy dy 1  y dx e 1  x
  • 47. ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey dx  ey dy dy 1  y dx e 1  x Exercise 13A; 1 to 4 ace etc, 6 to 8 ace, 10 to 12 ac Exercise 13B; 4, 7, 8 to 22 evens (not 18)

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