The document outlines differentiation rules: 1) The derivative of a constant function is 0. 2) The derivative of a function with respect to x multiplied by a constant k is the derivative of the function multiplied by k. 3) The derivative of a polynomial function is found by taking the derivative of each term. 4) The derivative of a function divided by x is the derivative of the function minus the function divided by x squared.

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=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =

6. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=

7. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=

8. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=

9. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=
( )f x kx=

10. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=
( )f x kx=
( ) ( )f x h k x h
kx kh
+ = +
= +

11. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=
( )f x kx=
( ) ( )f x h k x h
kx kh
+ = +
= +
( ) 0
lim
h
kx kh kx
f x
h→
+ −
′ =

12. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=
( )f x kx=
( ) ( )f x h k x h
kx kh
+ = +
= +
( ) 0
lim
h
kx kh kx
f x
h→
+ −
′ =
0
lim
h
kh
h→
=

13. Rules For Differentiation(1) y c=
( )f x c=
( )f x h c+ =
( ) 0
lim
h
c c
f x
h→
−
′ =
0
0
lim
h h→
=
0
lim0
0
h→
=
=
(2) y kx=
( )f x kx=
( ) ( )f x h k x h
kx kh
+ = +
= +
( ) 0
lim
h
kx kh kx
f x
h→
+ −
′ =
0
lim
h
kh
h→
=
0
lim
h
k
k
→
=
=

14. (3) n
y x=

15. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +

16. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +

17. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +

18. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K

19. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x=

20. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x= ( ) ( )
n
f x h x h+ = +

21. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x= ( ) ( )
n
f x h x h+ = +
( )
( )
0
lim
n n
h
x h x
f x
h→
+ −
′ =

22. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x= ( ) ( )
n
f x h x h+ = +
( )
( )
0
lim
n n
h
x h x
f x
h→
+ −
′ =
( ) ( ) ( ) ( ){ }1 2 2 1
0
lim
n n n n
h
x h x x h x h x x h x x
h
− − − −
→
+ − + + + + + + +
=
K

23. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x= ( ) ( )
n
f x h x h+ = +
( )
( )
0
lim
n n
h
x h x
f x
h→
+ −
′ =
( ) ( ) ( ) ( ){ }1 2 2 1
0
lim
n n n n
h
x h x x h x h x x h x x
h
− − − −
→
+ − + + + + + + +
=
K
( ) ( ) ( ){ }1 2 2 1
0
lim
n n n n
h
h x h x h x x h x x
h
− − − −
→
+ + + + + + +
=
K

24. (3) n
y x=
( ) ( )2 2
a b a b a b− = − +
( ) ( )3 3 2 2
a b a b a ab b− = − + +
( ) ( )4 4 3 2 2 3
a b a b a a b ab b− = − + + +
( ) ( )1 2 3 2 2 3 2 1n n n n n n n n
a b a b a a b a b a b ab b− − − − − −
− = − + + + + + +
M
K
( ) n
f x x= ( ) ( )
n
f x h x h+ = +
( )
( )
0
lim
n n
h
x h x
f x
h→
+ −
′ =
( ) ( ) ( ) ( ){ }1 2 2 1
0
lim
n n n n
h
x h x x h x h x x h x x
h
− − − −
→
+ − + + + + + + +
=
K
( ) ( ) ( ){ }1 2 2 1
0
lim
n n n n
h
h x h x h x x h x x
h
− − − −
→
+ + + + + + +
=
K
( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
x h x h x x h x x
− − − −
→
= + + + + + + +K

25. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K

26. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K

27. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=

28. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=

29. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=

30. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+

31. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =

32. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+

33. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+

34. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+

35. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+
2
1
x
−
=

36. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+
2
1
x
−
=
Note:

37. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+
2
1
x
−
=
Note:
( ) 1
f x x−
=

38. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+
2
1
x
−
=
Note:
( ) 1
f x x−
=
( ) 2
f x x−
′ = −

39. ( ) ( ) ( ) ( )
1 2 2 1
0
lim
n n n n
h
f x x h x h x x h x x
− − − −
→
′ = + + + + + + +K
1 1 1 1n n n n
x x x x− − − −
= + + + +K
1n
nx −
=
1
(4) y
x
=
( )
1
f x
x
=
( )
1
f x h
x h
+ =
+
( ) 0
1 1
lim
h
x h xf x
h→
−
+′ =
( )0
lim
h
x x h
hx x h→
− −
=
+
( )0
lim
h
h
hx x h→
−
=
+
( )0
1
lim
h x x h→
−
=
+
2
1
x
−
=
Note:
( ) 1
f x x−
=
( ) 2
f x x−
′ = −
2
1
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=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =

44. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +

45. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +

46. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +

47. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +

48. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+

49. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+
1
2 x
=

50. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+
1
2 x
=
Note:

51. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+
1
2 x
=
Note:
( )
1
2
f x x=

52. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+
1
2 x
=
Note:
( )
1
2
f x x=
( )
1
2
1
2
f x x
−
′ =

53. (5) y x=
( )f x x=
( )f x h x h+ = + ( ) 0
lim
h
x h x
f x
h→
+ −
′ =
x h x
x h x
+ +
×
+ +
( )0
lim
h
x h x
h x h x→
+ −
=
+ +
( )0
lim
h
h
h x h x→
=
+ +
( )0
1
lim
h
x h x→
=
+ +
1
x x
=
+
1
2 x
=
Note:
( )
1
2
f x x=
( )
1
2
1
2
f x x
−
′ =
1
2 x
=

54. ( )e.g. 7i y =

55. ( )e.g. 7i y =
0
dy
dx
=

56. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=

57. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=

58. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=

59. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=

60. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +

61. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +

62. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +

63. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +

64. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +

65. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +

66. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +

67. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −

68. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −

69. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=

70. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=

71. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=

72. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=

73. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=
( ) ( )
( )
3
If 3,
find 2
viii f x x
f
= −
′

74. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=
( ) ( )
( )
3
If 3,
find 2
viii f x x
f
= −
′
( ) 3
3f x x= −

75. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=
( ) ( )
( )
3
If 3,
find 2
viii f x x
f
= −
′
( ) 3
3f x x= −
( ) 2
3f x x′ =

76. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=
( ) ( )
( )
3
If 3,
find 2
viii f x x
f
= −
′
( ) 3
3f x x= −
( ) 2
3f x x′ =
( ) ( )
2
2 3 2f ′ =

77. ( )e.g. 7i y =
0
dy
dx
=
( ) 37ii y x=
37
dy
dx
=
( ) 10
iii y x=
9
10
dy
x
dx
=
( ) 2
3 6 2iv y x x= + +
6 6
dy
x
dx
= +
( ) ( )
2
2 1v y x= +
2
4 4 1x x= + +
8 4
dy
x
dx
= +
( ) 2
1
3vi y x
x
= +
2
3x x−
= +
3
3 2
dy
x
dx
−
= −
3
2
3
x
= −
( ) 2
vii y x x=
5
2
x=
3
2
5
2
dy
x
dx
=
5
2
x x=
( ) ( )
( )
3
If 3,
find 2
viii f x x
f
= −
′
( ) 3
3f x x= −
( ) 2
3f x x′ =
( ) ( )
2
2 3 2f ′ =
12=

78. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +

79. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +

80. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −

81. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −

82. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=

83. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =

84. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −

85. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −

86. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =

87. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −

88. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −

89. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −
2
3 12
dy
x
dx
= −

90. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −
2
3 12
dy
x
dx
= −
tangents are horizontal when 0
dy
dx
=

91. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −
2
3 12
dy
x
dx
= −
tangents are horizontal when 0
dy
dx
=
2
i.e. 3 12 0x − =

92. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −
2
3 12
dy
x
dx
= −
tangents are horizontal when 0
dy
dx
=
2
i.e. 3 12 0x − =
2
4
2
x
x
=
= ±

93. ( )
( )
3 2
Find the equation of the tangent to the curve 5 6 2
at the point 1,1
xix y x x= − +
3 2
5 6 2y x x= − +
2
15 12
dy
x x
dx
= −
( ) ( )
2
when 1, 15 1 12 1
dy
x
dx
= = −
3=
required slope 3∴ =
( )1 3 1y x− = −
1 3 3y x− = −
3 2 0x y− − =
( ) 3
Find the points on the curve 12 where the tangents
are horizontal
x y x x= −
3
12y x x= −
2
3 12
dy
x
dx
= −
tangents are horizontal when 0
dy
dx
=
2
i.e. 3 12 0x − =
2
4
2
x
x
=
= ±
( ) ( )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
x
( )y f x=

96. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangent

97. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal

98. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +

99. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +

100. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −

101. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −

102. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −
21=

103. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −
21=
1
required slope
21
∴ = −

104. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −
21=
1
required slope
21
∴ = −
( )
1
29 3
21
y x− = − −

105. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −
21=
1
required slope
21
∴ = −
( )
1
29 3
21
y x− = − −
21 609 3y x− = − +

106. A normal is a line perpendicular to the tangent at the point of contact
y
x
( )y f x=
tangentnormal
( )
( )
2
Find the equation of the normal to the curve 4 3 2 at
the point 3,29
xi y x x= − +
2
4 3 2y x x= − +
8 3
dy
x
dx
= −
( )when 3, 8 3 3
dy
x
dx
= = −
21=
1
required slope
21
∴ = −
( )
1
29 3
21
y x− = − −
21 609 3y x− = − +
21 612 0x y+ − =

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