Calculus and Analytical Geometry Lecture # 8 MTH 104.

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


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


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


dxxdf

cdxxcfd

dxdy )())((

dxxd )4( 3

dxxd )(4 3

Examples

213 1234 xx


1112

1212

x

xdxd

xdxd

1

2

1

x

xdxd

xdxd

xdxd


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


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


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


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


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


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


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


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

Calculus and Analytical Geometry Lecture # 8 MTH 104.

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