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12 x1 t02 01 differentiating exponentials (2014)

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Exponentials
Exponentials y x
Exponentials y y  ax a  1 1 x
Exponentials ya 0  a  1 x y y  ax a  1 1 x
Exponentials ya 0  a  1 x y 1 y  ax a  1  y  ax   0  a  1    x
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
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
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
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
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
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
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)
Differentiating Exponentials
Differentiating Exponentials y  e f x
Differentiating Exponentials y  e f x dy  f  x e f  x  dx
Differentiating Exponentials y  e f x dy  f  x e f  x  dx y  a f x
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
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
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
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
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
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
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
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
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
(v) y  3 x 2 e 4 x
(v) y  3 x 2 e 4 x dy  3 x 2 4e 4 x   e 4 x 6 x  dx
(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
(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
(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
(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 
(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 
(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 
(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 
(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
(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
(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
(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
(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
ix  y  4 x2
ix  y  4 x2 dy x2  2 xlog 4 4 dx
ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x
ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey
ix  y  4 x2 dy x2  2 xlog 4 4 dx  x  y  log x x  ey dx  ey dy
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
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
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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12 x1 t02 01 differentiating exponentials (2014)

  • 1.
  • 2.
  • 3.
  • 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.
  • 14.
  • 15.
    Differentiating Exponentials y  ef x dy  f  x e f  x  dx
  • 16.
    Differentiating Exponentials y  ef x dy  f  x e f  x  dx y  a f x
  • 17.
    Differentiating Exponentials y  ef 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  ef 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  ef 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  ef 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  ef 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  ef 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  ef 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  ef 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  ef 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)