1. Rules For Differentiation

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  c
f  x  h  c

5. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c

6. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h

7. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0

8. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx

9. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx
f  x   kx

10. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx
f  x   kx
f  x  h  k  x  h
 kx  kh

11. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx
kx  kh  kx
f  x   kx f   x   lim
h0 h
f  x  h  k  x  h
 kx  kh

12. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx
kx  kh  kx
f  x   kx f   x   lim
h0 h
f  x  h  k  x  h kh
 lim
 kx  kh h0 h

13. Rules For Differentiation
(1) y  c
cc
f  x  c f   x   lim
h0 h
f  x  h  c 0
 lim
h 0 h
 lim 0
h0
0
(2) y  kx
kx  kh  kx
f  x   kx f   x   lim
h0 h
f  x  h  k  x  h kh
 lim
 kx  kh h0 h
 lim k
h0
k

14. (3) y  x n

15. (3) y  x n
a 2  b 2   a  b  a  b 

16. (3) y  x n
a 2  b 2   a  b  a  b 
a 3  b3   a  b   a 2  ab  b 2 

17. (3) y  x n
a 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 n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 

19. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn

20. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn f  x  h   x  h
n

21. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn f  x  h   x  h
n
 x  h   xn
n
f   x   lim
h0 h

22. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn f  x  h   x  h
n
 x  h   xn
n
f   x   lim
h0 h
 lim

 x  h  x   x  h    x  h  x     x  h  x n2  x n1
n 1 n2

h0 h

23. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn f  x  h   x  h
n
 x  h   xn
n
f   x   lim
h0 h
 lim

 x  h  x   x  h    x  h  x     x  h  x n2  x n1
n 1 n2

h0 h
 lim
 h  x  h    x  h  x     x  h  x n2  x n1
n 1 n2

h0 h

24. (3) y  x n
a 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 n1  a n2b  a n3b 2    a 2b n3  ab n2  b n1 
f  x   xn f  x  h   x  h
n
 x  h   xn
n
f   x   lim
h0 h
 lim

 x  h  x   x  h    x  h  x     x  h  x n2  x n1
n 1 n2

h0 h
 lim

h  x  h    x  h  x     x  h  x n2  x n1
n 1 n2

h0 h
 lim  x  h    x  h  x     x  h  x n2  x n1
n 1 n2
h0

25. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0

26. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1

27. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1

28. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x

29. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x
1
f  x 
x

30. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x
1
f  x 
x
1
f  x  h 
xh

31. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1 
f  x  f   x   lim x  h x
x h0 h
1
f  x  h 
xh

32. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1 
f  x  f   x   lim x  h x
x h0 h
1 xxh
f  x  h   lim
xh h0 hx  x  h 

33. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1 
f  x  f   x   lim x  h x
x h0 h
1 xxh
f  x  h   lim
xh h0 hx  x  h 
h
 lim
h0 hx  x  h 

34. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1 
f  x  f   x   lim x  h x
x h0 h
1 xxh
f  x  h   lim
xh h0 hx  x  h 
h
 lim
h0 hx  x  h 
1
 lim
h0 x  x  h 

35. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1 
f  x  f   x   lim x  h x
x h0 h
1 xxh
f  x  h   lim
xh h0 hx  x  h 
h
 lim
h0 hx  x  h 
1
 lim
h0 x  x  h 
1
 2
x

36. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1  Note:
f  x  f   x   lim x  h x
x h0 h
1 xxh
f  x  h   lim
xh h0 hx  x  h 
h
 lim
h0 hx  x  h 
1
 lim
h0 x  x  h 
1
 2
x

37. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1  Note:
f  x  f   x   lim x  h x
x h0 h f  x   x 1
1 xxh
f  x  h   lim
xh h0 hx  x  h 
h
 lim
h0 hx  x  h 
1
 lim
h0 x  x  h 
1
 2
x

38. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1  Note:
f  x  f   x   lim x  h x
x h0 h f  x   x 1
1 xxh
f  x  h   lim
xh h0 hx  x  h  f   x    x 2
h
 lim
h0 hx  x  h 
1
 lim
h0 x  x  h 
1
 2
x

39. f   x   lim  x  h    x  h x     x  h  x n2  x n1
n 1 n2
h0
 x n1  x n1    x n1  x n1
 nx n1
1
(4) y 
x 1 1
1  Note:
f  x  f   x   lim x  h x
x h0 h f  x   x 1
1 xxh
f  x  h   lim
xh h0 hx  x  h  f   x    x 2
h 1
 lim  2
h0 hx  x  h  x
1
 lim
h0 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  x
xh x
f  x  h  x  h f   x   lim
h 0 h

44. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x

45. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 

46. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x 

47. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x 
1
 lim
h0
 xh x 

48. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x 
1
 lim
h0
 xh x 
1

x x

49. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x 
1
 lim
h0
 xh x 
1

x x
1

2 x

50. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x  Note:
1
 lim
h0
 xh x 
1

x x
1

2 x

51. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x  Note:
1
1 f  x  x 2
 lim
h0
 xh x 
1

x x
1

2 x

52. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x  Note:
1
1 f  x  x
2
 lim
h0
 xh x  1 1
f  x  x 2
1 2

x x
1

2 x

53. (5) y  x
f  x  x
xh x xh x
f  x  h  x  h f   x   lim 
h 0 h xh x
xhx
 lim
h0
h  xh x 
h
 lim
h0
h  xh x  Note:
1
1 f  x  x
2
 lim
h0
 xh x  1 1
f  x  x 2
1 2
 1
x x 
2 x
1

2 x

54. e.g.  i  y  7

55. e.g.  i  y  7
dy
0
dx

56. e.g.  i  y  7
dy
0
dx
 ii  y  37 x

57. e.g.  i  y  7
dy
0
dx
 ii  y  37 x
dy
 37
dx

58. e.g.  i  y  7
dy
0
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10

59. e.g.  i  y  7
dy
0
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx

60. e.g.  i  y  7
dy
0
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2

61. e.g.  i  y  7
dy
0
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

62. e.g.  i  y  7  v  y   2 x  1
2
dy
0
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

63. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

64. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x
dy
 37
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

65. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

66. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
 iii  y  x10
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

67. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
dy
 3  2 x 3
 iii  y  x10 dx
dy
 10 x 9
dx
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

68. e.g.  i  y  7  v  y   2 x  1
2
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
dy
 3  2 x 3
 iii  y  x10 dx
dy 2
 10 x 9  3 3
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

69. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
dx dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
dy
 3  2 x 3
 iii  y  x10 dx
dy 2
 10 x 9  3 3
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

70. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4
dx
 ii  y  37 x 1
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
dy
 3  2 x 3
 iii  y  x10 dx
dy 2
 10 x 9  3 3
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

71. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2
 37 x
dx  3 x  x 2
dy
 3  2 x 3
 iii  y  x10 dx
dy 2
 10 x 9  3 3
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

72. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  3 x  x 2 2
dy
 3  2 x 3
 iii  y  x10 dx
dy 2
 10 x 9  3 3
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

73. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  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
2 find f   2 
dx x
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

74. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  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
2 find f   2 
dx x
f  x   x3  3
 iv  y  3x 2  6 x  2
dy
 6x  6
dx

75. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  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
2 find f   2 
dx x
f  x   x3  3
 iv  y  3x 2  6 x  2 f   x   3x 2
dy
 6x  6
dx

76. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  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
2 find f   2 
dx x
f  x   x3  3
 iv  y  3x 2  6 x  2 f   x   3x 2
dy
 6x  6
f   2  3 2
2
dx

77. e.g.  i  y  7  v  y   2 x  1
2
 vii  y  x 2 x
dy
0  4x2  4 x  1
5
dx x 2
dy
 8x  4 dy 5 3
dx  x2
 ii  y  37 x 1 dx 2
dy  vi  y  3x  2 5
 37 x  x x
dx  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
2 find f   2 
dx x
f  x   x3  3
 iv  y  3x 2  6 x  2 f   x   3x 2
dy
 6x  6
f   2  3 2
2
dx
 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 1,1
y  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 1,1
y  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 1,1
y  5 x3  6 x 2  2
dy
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2
dx

82.  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
dy
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2
dx
3

83.  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
dy
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2
dx
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
dy
when x  1,  15 1  12 1
2
dx
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
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2
dx
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
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are 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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are 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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are horizontal
y  x 3  12 x
dy
 3 x 2  12
dx

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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y  x 3  12 x dx
dy
 3 x 2  12
dx

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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y  x 3  12 x dx
i.e. 3 x  12  0
2
dy
 3 x 2  12
dx

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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y  x 3  12 x dx
i.e. 3 x  12  0
2
dy
 3 x 2  12 x2  4
dx
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  1
dy y  1  3x  3
 15 x 2  12 x
dx
dy
when x  1,  15 1  12 1
2 3x  y  2  0
dx
3
 required slope  3
 x  Find the points on the curve y  x3  12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y  x 3  12 x dx
i.e. 3 x  12  0
2
dy
 3 x 2  12 x2  4
dx
x  2
 tangents are horizontal at  2,16  and  2, 16 

94. A normal is a line perpendicular to the tangent at the point of contact

95. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x

96. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
tangent

97. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent

98. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 

99. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
y  4 x 2  3x  2

100. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
y  4 x 2  3x  2
dy
 8x  3
dx

101. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
y  4 x 2  3x  2
dy
 8x  3
dx
dy
when x  3,  8  3  3
dx

102. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
y  4 x 2  3x  2
dy
 8x  3
dx
dy
when x  3,  8  3  3
dx
 21

103. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
y  4 x 2  3x  2
dy
 8x  3
dx
dy
when x  3,  8  3  3
dx
 21
1
 required slope  
21

104. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
1
y  4 x  3x  2
2
y  29    x  3
dy 21
 8x  3
dx
dy
when x  3,  8  3  3
dx
 21
1
 required slope  
21

105. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
1
y  4 x  3x  2
2
y  29    x  3
dy 21
 8x  3 21 y  609   x  3
dx
dy
when x  3,  8  3  3
dx
 21
1
 required slope  
21

106. A normal is a line perpendicular to the tangent at the point of contact
y
y  f  x
x
normal tangent
 xi  Find the equation of the normal to the curve y  4 x 2  3 x  2 at
the point  3, 29 
1
y  4 x  3x  2
2
y  29    x  3
dy 21
 8x  3 21 y  609   x  3
dx
dy
when x  3,  8  3  3 x  21 y  612  0
dx
 21
1
 required slope  
21

107. Exercise 7C; 1ace etc, 2ace etc, 3ace etc, 4bd, 5bdfh,
8bd, 9bd, 10ac, 12, 13b, 16, 21
Exercise 7D; 2ac, 3bd, 4ace, 6c, 7b, 11a, 13aei, 18, 22