Calculus and Analytical Geometry

Lecture # 8

MTH 104

Techniques of differentiation

1. Constant Function Rule:The derivative of a constant function is zero. y = f(x) = cwhere c is a constant

.0

dxdc

dxxdf

dxdy

Examples

,0)1(

dxd ,0

)5( dxd .0

)2( dx

d

Techniques of differentiation

nxxfy )(

xxxdxd

22 122

1 nn nxxdxd

2. Power Rule:Let , where the dependant

variable x is raised to a constant value, the power n, then

5xdxd

45x

Examples

Techniques of differentiation

2

1

xdxd

xxx

21

21

21

2

11

2

1

7xdxd 817 77 xx

78 8xxdxd

3. Constant Multiplied by a Function Rule:Let y be equal to the product of a constant c and some function f(x), such that y = cf(x) then

Techniques of differentiation

dxxdf

cdxxcfd

dxdy )())((

dxxd )4( 3

dxxd )(4 3

Examples

213 1234 xx

Techniques of differentiation

1112

1212

x

xdxd

xdxd

1

2

1

x

xdxd

xdxd

xdxd

Techniques of differentiation

xgdxd

xfdxd

xgxfdxd

dxdy

962 xxdxd 962 x

dxd

xdxd

4. Sum (Difference) Rule:Let y be the sum (difference) of two functions (differentiable) f(x) and g(x).

y = f(x) + g(x),

then

Examples

)9(62 195 xx105 912 xx

Techniques of differentiation

x

x

x

xdxd

dxd

xdxd

1

21

-2

dxd

2-0

2121

2

1

2

1

Techniques of differentiation.9523 38 xxxy

9523 38 xxxdxd

dxdy

9523 38

dxd

xdxd

xdxd

xdxd

Example Find dy/dx if

solution

0)1(53283 27 xx

5624 27 xx

Techniques of differentiation

433 xxy

33

43

2

3

x

xxdxd

dxdy

Example At what points, if any does the graph of

have a horizontal tangent line? solution

1

1

01

033 0

2

2

2

x

x

x

xdxdy

Slope of horizontal line is zero that is dy/dx=0

Techniques of differentiation

4. Product Rule:Let y = f(x).g(x), where f(x) and g(x) are two

differentiable functions of the variable x. Then

xfdxd

xgxgdxd

xfdxdy

xgxfdd

dxdy

Techniques of differentiation

xxxy 32 714

xxxdxd

dxdy 32 714

147714 2332 xdxd

xxxxdxd

x

Example Find dy/dx, if

solution

xxxxx 8712114 322

19140

85612148424

24224

xx

xxxxx

Techniques of differentiation

5. Quotient Rule:Let y = f(x)/g(x), where f(x) and g(x) are two differentiable functions of the variable x and g(x) ≠ 0. Then

2

xg

xgdxd

xfxfdxd

xg

dxdy

xgxf

dxd

dxdy

Techniques of differentiation

514

2

xx

y

514

2xx

dxd

dxdy

22

22

5

5)14(145

x

xdxd

xxdxd

x

Example Find dy/dx if

solution Derivative of numerator

Derivative of denominator

Techniques of differentiation

22

2

52)14(45

xxxx

22

22

528204

xxxx

22

2

52024

xxx

Higher order derivatives

nfffff 4,,,If y=f(x) then

xfdxd

dxyd

y

xfdxd

xfdxd

dxd

dxyd

y

xfdxd

dxdy

y

3

3

3

3

2

2

2

2

Higher order derivatives

A general nth order derivative

xf

dxyd

xfxfdxd

dxyd

n

n

n

n

n

n

n

n

and

Example

constants. are

,, where if Find 24 cbacbxaxyy

cbxaxdxd

dxdy

y 24

cdxd

bxdxd

axdxd 24

Solution

bxax 24 3 bxax

dxd

dxyd

y 24 3

2

2

First Orderderivat

ive

Second orderderivative

)2()4(2

2

3

2

2

2

2

bxdxd

axdxd

dxyd

y

bax 212 2

ax

baxdxd

dxyd

y

12

212 2

3

3

3

3

Third order derivative

Example Find 25

1x

2

2

46 where xxydx

yd

xx

xx

xxdxd

dxdy

830

2456

46

4

4

25

xxdxd

dxyd

830 4

2

2

Solution

8120 3 x

1128)1(1201

2

2

x

dx

yd

Derivative of trigonometric functions

xxdxd

cossin .1

xcoxdxd

sin .2

xxdxd

2sectan .3

xxxdxd

tansecsec .5

xxxdxd

cotcsccsc .4

xxdxd

2csccot .6

Example . find cossin2

2

dxyd

xxy

xxdxd

dxdy

cossin

xdxd

xxdxd

x sincoscossin

Solution

)(coscos)sin(sin xxxx

xx 22 cossin

xdxd

xdxd

dxyd

22

2

2

cossin

xx

xxxx

cossin4

cossin2cossin2

Example

xyy

xxy

cos2

osolution t a is sin that Show

1 cos2 xyy

xxxy r.t w.sin times twoatingDifferenti

solution

xxx

xxdxd

dxdy

y

sincos

sin

xdxd

xxdxd

xxxdxd

y sincossincos

xxx

xxxxy

cos2sin

coscossin

yy and

xxxxxx cos2sincos2sin

Substituting the valuse of into (1)

xx cos2cos2 L.H.S=R.H.S

Example Given that xxxf tansin)( 2 24

fshow that

xxxf tansin)( 2

xxxf tansin)( 2xx 22 secsin )(xf ))(cos(sin2 tan xxx

xx

22

cos

1sin

x2tan x2sin2

4f

4tan2

4

sin2 2

12

2

12

2

1

))(cos(sin2

cos

sin xx

x

x