9 differentiation

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Chapter 9: DIFFERENTIATION

Determine the first derivative of the function. (Week 1)

1. y = x 3 2. y = x 4

[4x3]

3. y = x 5

[5x4]

4. y = x 7

[7x6]

5. y = 2x3 6. y = 3x 4

[12x3]

7. y = 5x 3

[15x2]

8. y = 10x 2

[20x]

9. y = – 2x3 10. y = – 5 x 4

[ -20x3]

11. y = – 8 x 5

[-40x4 ]

12. y = – 12 x 2

[-24x]13. f(x) = x-–2 14. f(x) = x -1

21

x[- ]

15. f(x) = x -5

65

x[- ]

16. f(x) = x -3

26

x[- ]

17. f(x) = 3 x -2 18. f(x) = 4x -1

24

x[- ]

19. f(x) = 2x -4

58

x[- ]

20. f(x) = 6x -1

26

x[- ]

21. y =2

1x4 22. y =

2

3x4

[ 6x3]

23. y =3

1x 6

[2x5 ]

24. y =6

1 x – 3

41

2x[ ]

25. y =3

2 x3 26. y =

3

2 x – 6

74

x[ ]

27. f(x) =x2

5

25

2x[- ]

28. f(x) =63

4

x

78

x[- ]


29. y = x2 + 3x +4 30. y = x2 +4x +3

[2x+4]

31. y = x2 + 5x +6

[2x+5]

32. y = x2 - 3x +4 33. y = x2 -4x +3

[2x-4]

34. y = x2 - 5x +6

[2x-5]

35. y = x3 + 4x2 + 5 36. y = x3 +5x2 +7

[3x2 +10x]

37. y = x3 + 6x2 + 8

[3x2 + 12x]

38. y = x3 - 3x2 -6 39. y = x3 -5x2 -7

[3x2-10x]

40. y = x3 - 6x2 – 8

[3x2 -12x]

41. y = x(x + 5) 42. y = x(x - 6)

[2x-6]

43. y = x2(x + 5)

[ 3x2 + 10x]

44. y = (x+1)(x + 5) 45. y = (x+1)(x – 6)

[2x-5]

46. y = (x2 +1)(x - 4)

[3x2 -8x+1]

47. y = (x+3)2 48. y = (x+4)2

[2(x+4)]

49. y = (3x+1)2

[6(3x+1)]

50. y = x(x+3)2 51. y = x(x+4)2

[3x2 + 16x+16]

52. y = x(3x+1)2

[27x2 +12x+1]


Product Rule & Quotient Rule (Week 2)

Determine the first derivative of the function using the product rule.

1. y= x ( x3+1) 2. y= x ( x4+2)

[ 5x4+2]

3. y= ( x 5+1)x

[ 6x5+1]

4. y= ( x3-1)x

[4x3 -1]

5. y= 2x2 ( x3+1)6.

7. y= 3x2 ( x3-1)

[15x4-6x]

8. y= ( x3-1)(5x2)

[25x4-10x]

9. y= ( x3-1)(-4x2)

[-20x4+8x]

10. f(x)= (x+1) ( x3+1) 11. f(x)= (x-1) ( 1+x3)

[4 x3-3x2+1]

12. f(x)= (1-x) ( x3+2)

[-4x3+3x2-2 ]

13. f(x)= (2-x) ( x3+3)

[-4x3+6x2-3]


14. y= (x+1) (2

x+1) 15. y= (x-1) (

3

x+1)

[3

2

3

2x ]

16. y= (2x+1) (2

x+1)

[2

122 x ]

17. y= (3-2x) ( 2-2

x)

[2

152 x ]

18. y= (x2+1) (2

x+1) 19. y= (x4+1) (

3

x+1)

[3

14

3

5 34 xx ]

20. y= (2+x2) (4

x-1)

[ ]2

12

4

3 2 xx

21. y= (3-x3) (3

x+1)

[ - 133

4 23 xx ]


Determine the first derivative of the function using the quotient rule.

Example :1

x

xy 1.

32

5

x

xy

215

(2x+3)

2. y =54

3

x

x

215

(4x+5)

3. y =72

6

x

x

242

(2x-7)-


4. y =23

45

x

x

222

(3x-2)-

5. y =x

x

1

41

25

(1+x)-

6.x

xy

21

1

21

(1-2x)

7. y =3

2

x

x

xx-3


8. y =x

x

51

3 2

2

26x-15x

(1-5x)

9. y =10

42

3

x

x

4 2

2 24x +120x

(x +10)

10.

2

3

5x - 3xy =

8 + x

4 3

3 2-5x +6x +80x-24

(8+x )

11. y =45

13

2

x

x

4 2

3 2-5x +15x -8x

(5x -4)


Chain Rule (Week 3)

Determine the first derivative of composite function using chain rule.

1.2)2( xy 2.

4)3( xy

[4(x+3)3]

3.5)2( xy

[5(x+2)4]

4.3)8( xy

[3(x+8)2]

5.2)23( xy 6.

4)32( xy

[8(2x+3)3]

7.5)24( xy

[20(4x+2)4]

8.3)85( xy

[15(5x+8)2]

9.2)2(3 xy 10.

4)2(5 xy

[20(x+2)3]

11.5)24(3 xy

[60(4x+2)4]

12.3)82(2 xy

[12(2x+8)3]

13.2)2(

2

xy 14.

4)2(

5

xy

520

(x+2)[- ]

15.5)2(

3

xy

615

(x+2)[- ]

16.3)8(

2

xy

46

(x+8)[- ]

17.2)2(5

2

xy 18.

3)5(4

3

xy

49

4(x+5)[- ]

19.6)32(5

4

xy

724

5(2x-3)[- ]

20.4)43(2

5

xy

510

(3x-4)[ ]


Equations of Tangent and a Normal

Determine the Gradient of a Tangent and a Normal at a point on a Curve.

Example 1 : Find the gradient of the tangent to

the curve 5732 23 xxxy

at the point (-2,5).

Example: Given )32()( xxxf and the

gradient.

1. Given that the equation of a parabola is2241 xxy , find the gradient of the

tangent to the curve at the point (-1,-3)

8Tm

2. Find the gradient of the tangent to the curve

32 xxy at the point (3,6).

7Tm

3. Given that the gradient of the tangent at point P

on the curve 252 xy is – 4, find the

coordinates the point P.

P(2 , 1)

4. Given2

4)(

xxxf and the gradient of

tangent is 28. Find the value of x.

32x


Determine the equations of a Tangent and a Normal at a point on a Curve.

Example 1 :Find the equation of the tangent at the point (2,7)

on the curve 53 2 xy

Example 2 :Find the equation of the normal at the point x = 1

on the curve2324 xxy

1. Find the equation of the tangent at the point (1,9)

on the curve 252 xy

2112 xy

2. Find the equation of the tangent to the curve

112 xxy at the point where its x-

coordinate is -1.

33 xy

3. Find the equation of the normal to the curve

232 2 xxy at the point where its

x- coordinate is 2.

0225 yx

4. Find the gradient of the curve32

4

xy at the

point (-2,-4) and hence determine the equation ofthe normal passing through that point.

0308;8 yxmT


Second Derivatives (Week 4)Example :

Given that 5223)( xxf , find f ”(x) .

Hence, determine the value of f ”(1)Solution:

Example :

Given that 42 3)( xxf , find f ”(x) .

Hence, determine the value of f ”(2)Solution:

1. Given that 3642 23 xxxy , find2

2

dx

yd

12x + 8

2. Given that xxxf 3402)( 2 , find f”(x).

160 – 36x

3. Given that5)14()( xxf , find f ”(0)

–320

4. Given that 22 13 ts ,calculate the value of

2

2

dt

sdwhen t =

2

1.

–3


Turning Points (Minimum and Maximum Points)

Example :Find the turning points of the curve

318122 23 xxxy and determine whether

each of them is a maximum or a minimum point.

1. Find the coordinates of two turning points on the

curve 32 xxy

(1 , –2) and (–1 , 2)


2. Given 523

2 23 xxy is an equation of a

curve, find the coordinates of the turning pointsof the curve and determine whether each of theturning point is a maximum or minimum point.

min. point = (0 , –5) ; max. point = ),2( 323

3. Determine the coordinates of the minimum point

of 442 xxy .

(2 , 0)


Rates of Change (Week 5)

1. Given that xxy 23 2 and x is increasing at a

constant rate of 2 unit per second, find the rate ofchange of y when x = 4 unit.

2. Given that xxy 24 and x is increasing at a

constant rate of 4 unit per second, find the rate ofchange of y when x = 0.5 unit.

12 unit s –1

3. Given thatx

xv1

9 and x is increasing at a

constant rate of 3 unit per second, find the rate ofchange of v when x = 1 unit.

30 unit s –1

4. Two variables, x and y, are related by the

equation .2

3x

xy Given that y increases at a

constant rate of 4 unit per second, find the rate ofchange of x when x =2.

58 unit s –1


5. The area of a circle of radius r cm increases at aconstant rate of 10 cm2 per second. Find the rateof change of r when r = 2 cm. ( Use п = 3.142 )

6. The area of a circle of radius r cm increases at aconstant rate of 16 cm2 per second. Find the rateof change of r when r = 3 cm. ( Use п = 3.142 )

0.8487 cm s –1

7. The volume of a sphere of radius r cm increasesat a constant rate of 20 cm3 per second. Find therate of change of r when r = 1 cm. ( Use п = 3.142 )

1.591 cm s –1

8. The volume of water , V cm³, in a container is

given by ,83

1 3 hhV where h cm is the

height of the water in the container. Water ispoured into the container at the rate of

.scm10 -13 Find the rate of change of the height

of water, in ,scm -13at the instant when its

height is 2 cm.

65 cm s –1


Example :

The above figure shows a cube of volume 729cm³. If the water level in the cube, h cm, is

increasing at the rate of 0.8 cm s 1 , find the rateof increase of the volume of water.

Solution :

Let each side of the cube be x cm.Volume of the cube = 729 cm³

x³ = 729x = 9

Rate of change of the volume of water,

1364.8

0.881

scm

dt

dh

dh

dV

dt

dV

Hence, the rate of increase of the volume of

water is 64.8 cm³ s 1 .

9. A spherical air bubble is formed at the base of apond. When the bubble moves to the surface ofthe water, it expands. If the radius of the bubble

is expanding at the rate of 0.05 cm s 1 , find therate at which the volume of the bubble isincreasing when its radius is 2 cm.

8.0 cm3 s –1

10. If the radius of a circle is decreasing at the

rate of 0.2 cm s 1 , find the rate of decreaseof the area of the circle when its radius is 3cm.

2.1 cm2 s –1

h cm

9 cm

9 cm

9 cm

h cm

Chain rule

V = 9 x 9 x h = 81h

=81

=rate of increase of

the water level

= 0.8 cm s


11. The radius of a spherical balloon increases at therate of 0.5 cm s –1. Find the rate of change in thevolume when the radius is 15 cm.

450 cm3 s –1

12. The edge of a cube is decreasing at the rate of3 cm s –1. Find the rate of change in the volumewhen the volume is 64 cm3.

–144 cm3 s –1

13. Diagram 1 shows a conical container with adiameter of 60 cm and height of 40 cm. Water ispoured into the container at a constant rate of

1 000 -13 scm .

Calculate the rate of change of the radius of the waterlevel at the instant when the radius of the water is 6

cm.

(Use π = 3.142; volume of cone hr 2

3

1 )

6.631cm3 s –1

14. Oil is poured into an inverted right circular coneof base radius 6 cm and height 18 cm at the rate

of 2 -13 scm . Find the rate of increase of theheight of water level when the water level is 6 cmhigh. ( Use п = 3.142 )

0.1591 cm s –1

60 cm

Water40 cm

Diagram 1


15.

The above figure shows an inverted cone with aheight of 20 cm and a base-radius of 4 cm. Water is

poured into the cone at the rate of 5 cm³ s 1 but atthe same time, water is dripping out from the cone

due to a leakage a the rate of 1 cm³ s 1 .a. If the height and volume of the water at time

t s are h cm and V cm³ respectively, showthat

.75

1 3hV

b. Find the rate of increase of the water level inthe cone at the moment the water level 12cm.

( Use п = 3.142 )

(b) 0.2210 cm s –1

16.

The above diagram shows a solid which consists of acuboid with a square base of side 6x cm,surmounted by a pyramid of height 4x cm. Thevolume of the cuboid is 5832 cm³.

a. Show that the total surface area of the solid,

A cm², is given by .3888

96 2

xxA

b. If the value of x increasing at the rate 0.08

,scm -1find the rate of increase of the total

surface area of the solid at the instant x = 4.

42 cm2 s –1

4 cm

20 cm

h cm

B

H

F G

C

A

D

4x cm

6x cm

6x cm

V

E


Small Changes and Approximations (Week 6)

1. Given that xxy 42 , find the small

change in y when x increases from 2 to 2.01.

2. Given that xxy 32 , find the small

change in y when x increases from 6 to 6.01.

0.15

3. Given that xxy 22 , find the small change in

y when x decreases from 8 to 7.98.

–0.62

4. Given that xy 4 , finddx

dy. Hence, find the

small change in y when x increases from 4 to4.02.

02.0;2 yxdx

dy


5. Given the area of a rectangle , xxA 23 2 ,where x is the width, find the small change in thearea when the width decreases from 3 cm to 2.98cm.Answer :

6. A cuboid with square base has a total surface

area, xxA 43 2 , where x is the length of theside of the base. Find the small change in the totalsurface area when the length of the side of thebase decreases from 5 cm to 4.99 cm.

–0.26 cm2

7. The volume, V cm3 , of a cuboid with rectangular

base is given by xxxV 32 23 , where x cmis the width of the base. Find the small change inthe volume when the width increases from 4 cmto 4.05 cm.

1.75 cm3

8. In a pendulum of length x meters, the period T

seconds is given as10

2x

T . Finddx

dT.

Hence, find the small change in T when xincreases from 2.5 m to 2.6 m.

5010; T

xdxdT

second


Example :The height of a cylinder is three times its radius.Calculate the approximate increase in the totalsurface area of the cylinder if its radius increasesfrom 7 cm to 7.05 cm.

Solution :

Let the total surface area of the cylinder be Acm².

A = Sum of areas of the top and bottom circularsurface + Area of the curved surface.

2

2

2

8

322

22

r

rrr

rhrA

Approximate change in the total surface area isA

2cm6.5

05.0716

705.716

r

rdr

dAA

dr

dA

r

A

Hence, the approximate increase in the totalsurface area of the cylinder is 5.6π cm² .

9. A cube has side of 6 cm. If each of the sideof the cube decreases by 0.1 cm, find theapproximate decrease in the total surface areaof the cube.

7.2 cm2

10. The volume of a sphere increases from

.cm290tocm288 33 Calculate theapproximate increase in its radius.

721

cm

It is given thath=3r

New r (7.05)Minus old r(7)

Substitute r with theold value of r, i.e. 7


Given that5

4

xy , calculate the value of

dx

dyif

x = 2. Hence, estimate the values of

55 98.1

4(b)

03.2

4)a(

Solution :

16

520,2When

2020

44

6

6

6

5

5

xdx

dyx

xx

dx

dy

xx

y

xdx

dyy

yyya

original

original

)( new

.1165250

320

3

32

4

203.216

5

2

4

2

4

03.222

4,

2

4

03.2

4

5

5

555

xdx

dy

xandywherey

0.13125

160

1

32

4

298.116

5

2

4

1.98

4

(b)

55

xdx

dyyy

yyy

originalnew

originalnew

11. Given3

27

xy , find the value of

dx

dywhen

x = 3. Hence, estimate the value of .03.3

273

1dx

dy; 0.97

12. Given4

32

xy , find

dx

dy.

Hence, estimate the value of .99.1

324

5128

xdx

dy ; 2.04


13. Given thatx

y

2

20, find the approximate

change in x when y increases from 40 to 40.5.

1601

14. Given that ,52 xxy use differentiation to

find the small change in y when x increases from3 to 3.01. [3marks]

0.11

15. Given 3

3

4rv , use the differentiation method

to find the small change in v when r increasesfrom 3 to 3.01.

27.0

16. Given that ,5

3xy find the value of

dx

dywhen x

= 4. Hence, estimate the value of

(a) 302.4

5(b)

399.3

5

25615dx

dy; (a) 0.07930 ; (b) 0.07871


Maximum and Minimum Values of a Function

Example 1 :

Given )540(2 xxL , find the value of x for which

L is maximum. Hence, determine the maximum valueof L.

Solution:

27

10240

3

16540

3

16

max3

16

080

803

163080

3

16

01580,0

0)1580(

01580

0

3080

1580

540

)540(

2

max

2

2

2

2

2

2

2

2

32

2

L

Limisewillx

dx

Ld

dx

Ld

x

xxKnowing

xx

xx

dx

dLwhen

xdx

Ld

xxdx

dL

xxL

xxL

1. Given )225(2 xxL , find the value of x for

which L is maximum. Hence, determine themaximum value of L.

2715625

max325 , Lx

2. Given 7436 2 xxy , find the value of x or

which y is maximum. Hence, determine themaximum value of y.

88, max29 yx


Example 2 :

Given )16

(3 2

xxL , find the value of x for

which L is minimum. Hence, determine the minimumvalue of L.

Solution:

36

2

1623

min2

018

)2(966

2

8,0

486

486

0486

0

966

486

483

)16

(3

2

min

2

2

3

2

2

3

2

2

2

3

2

2

2

12

2

L

Limisewillx

dx

Ld

dx

Ld

x

xxKnowing

xx

xx

xx

dx

dLwhen

xdx

Ld

xxdx

dL

xxL

xxL

3. Given )128

(4 2

xxL , find the value of x for

which L is minimum. Hence, determine theminimum value of L.

192,4 min Lx

4. Given )283(2

1 2 xxy , find the value of x

for which y is minimum.Hence, determine the minimum value of y.

35

min34 , yx

5. Given that y=14x(5 – x), calculatea) the value of x when y is a maximumb) the maximum value of y

[3 marks]

(a) 2.5 ; (b) 87.5


6. The diagram below shows the skeleton of a wire. Box with a rectangular base of x by 3x and theheight of h.

Given the total length of the wire is 348 cm,(a) express h in terms of x,(b) show that the volume of the box, V cm3 , is given by V = 3x2 (87 – 4x),(c) find the stationary value of V, stating whether it is maximum or minimum value.

(a) h = 87 – 4x (b) – (c) maxV = 18291.75 cm3

7. The diagram below shows a solid cylinder with circular cross-section of radius r and the height of h.

Given the total surface area of the cylinder is 300 cm2 ,(a) express h in terms of r,(b) show that the volume of the cylinder, V cm3, is given by V = 150 r – п r 3 ,(c) determine the maximum volume of the cylinder when r varies.

(a) h = rr

2150

(b) – (c) 399or100 50

8. A length of wire 160 m is bent to form a sector OPQ, of a circle of centre O and radius r as shownin the diagram below.

(a) Show that (i) 2160

r

(ii) the area, A, of the sector isgiven by A = 80r – r 2

(b) Find the value of θ and r when the area is a maximum. (b) Determine the maximum area.

(b) r = 40 m , θ = 2 rad (c) 1600max A

x3x

h

r

h

r

θ


SPM Questions (Week 7&8)

1. The gradient function of a curve which passes through A (1, – 12) is 3x2 – 6x. Finda. the equation of the curve,

[3 marks]b. the coordinates of the turning points of the curve and determine whether each of the

turning points is a maximum or a minimum.[5 marks]

2. The point P lies on the curve y = (x – 5)2. It is given that the gradient of the normal at P is1

4 . Find the coordinates of P.

[3 marks]

3. It is given that 72

3y u , where u = 3x – 5. Find

dy

dxin terms of x.

[3 marks]

4. Given that y = 3x2 + x – 4 ,

a. find the value ofdy

dxwhen x = 1,

b. express the approximate change in y, when x changes from 1 to 1 + p, where p if asmall value.

[4 marks]

5. Given that

2

1( )

3 5h x

x

, evaluate "(1)h .

[4 marks]

6. The volume of water, V cm3, in a container is given by 318

3V h h , where h cm is the

height of the water in the container. Water is poured into the container at the rate of 10 cm3 s-1. Find the rate of change of water, in cm s-1, at the instant when its height is 2 cm.

[3 marks]7. Differentiate 3x2 (2x – 4)4 with respect to x.

[3 marks]

8. Two variables, x and y, are related by the equation2

3y xx

. Given that y increases at a

constant rate of 4 units per second, find the rate of change of x when x = 2.[3 marks]

9 differentiation

Documents

xdxl ddxl

total surface

quotient rule

turning points

water level

small change

totalsurface

approximate