7. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
8. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
9. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
f x kx
10. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
f x kx
f x h k x h
kx kh
11. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h
kx kh
12. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h kh
lim
kx kh h0 h
13. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h kh
lim
kx kh h0 h
lim k
h0
k
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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f x lim
h0 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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f 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 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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f 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 n1
n 1 n2
h0 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 n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f 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 n1
n 1 n2
h0 h
lim x h x h x x h x n2 x n1
n 1 n2
h0
25. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
26. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
27. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
28. f x lim x h x h x x h x n2 x n1
n 1 n2
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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x
1
f x
x
30. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x
1
f x
x
1
f x h
xh
31. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1
f x h
xh
32. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
33. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f 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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f 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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f 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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h
1 xxh
f 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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f 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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f x h lim
xh h0 hx x h f x x 2
h
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 n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f x h lim
xh h0 hx x h f x x 2
h 1
lim 2
h0 hx x h x
1
lim
h0 x x h
1
2
x
43. (5) y x
f x x
xh x
f x h x h f x lim
h 0 h
44. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
45. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
46. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
47. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
48. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
1
x x
49. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
1
x x
1
2 x
50. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
lim
h0
xh x
1
x x
1
2 x
51. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x
1
x x
1
2 x
52. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x 1 1
f x x 2
1 2
x x
1
2 x
53. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x 1 1
f x x 2
1 2
1
x x
2 x
1
2 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