derivative module 2 september 2011

8/12/2019 Derivative Module 2 September 2011

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30/09/2011 Techniques of differentiation 1

Computing Derivative

by using the techniques/rules ofdifferentiation.

basic functions.

MODULE 2


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Learning Objective

At the end of the module, students should beable to compute differentiation using suitabledifferentiation technique such as

power rule, product rule,

quotient rule, sum rule and difference rule constant multiple rule. basic functions.


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Introduction

Computing derivatives using definition canbe tedious, since the evaluation ofthe limit can be difficult.


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Fortunately, several formulas or rules havebeen developed in finding derivativeswithout having to use the definitiondirectly.


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DIFFERENTIATION TABLE andTECHNIQUES


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1. Derivative of a constant function

Givenf(x) = c, then

0)()( cdx

d

dx

dfxf


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1. Derivative of a constant function

Givenf(x) = c, then using definition

h

xfhxfxf

h

)()(lim)(

0

Proof:

00lim0

limlim000 hhh hh

cc


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2. The Power Rule

Givenf(x) =xn, then

1)( nnxxf


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Proof

h

xfhxfxf

h

)()(lim)(

0

f(x) =xn

h

xhxxf

nn

h

)(lim)(

0

Recall: )...)(( 1221 nnnnnn yxyyxxyxyx

h

xxhxxhxhxxhxxf

nnnn

h

)))(...)())(()(((lim)(

1221

0


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h

xxhxxhxhxhxf

nnnn

h

)))(...)())(((lim)(

1221

0

)))(...)()((lim)( 12210

nnnn

hxxhxxhxhxxf

)))(...)()((lim)( 12210

nnnn

hxxxxxxxf

)...()( 1111 nnnn xxxxxf n times

1)( nnxxf


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3. The constant multiple rule

Given a constant c and f(x), then

)()]([ xfdx

dcxcf

dx

d


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The derivatives of basic

trigonometric function

3.

2.

1.

xxdx

d

xxdx

d

xxdx

d

2sec)(tan

sin)(cos

cos)(sin

xxxdx

d

xxdx

d

xxxdx

d

cotcsc)(csc

csc)(cot

tansec)(sec

2

6.

5.

4.


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cos)(sin xxdx

dProve

h

xhxh

xfhxfxf

h

h

)sin()sin(lim

)()(lim)(

0

0

Proof:


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h

xx

hxxx

h

xxxh

xhxxf

h

h

h

h

cossinh)1(coshsinlim

cossinhsincoshsinlim

sincossinhcoshsinlim

)sin()sin(lim)(

0

0

0

0


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xxf

hx

hx

hh

cos)(

cossinhlim)1(coshsinlim00


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Derivative of Exponential Function


1. Given then,xby

dx

dybbx ln

2. If b = e xey

dx

dyeex ln

xe


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10ln1010.1

xx

dx

dy

y

3.0ln)3.0()3.0(.2 xx

dx

dyy


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Derivative of Logarithmic Function


bxx

dx

db

ln

1)(log1.

0,1

)(ln xx

xdx

d2.

Also known asNatural Logarithm


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30/09/2011 Techniques of differentiation 1930/09/2011Techniques of differentiation 19

)(log xdx

db bx ln

1

Proof

)(log xdx

db )

ln

ln(

b

x

dx

d

)(lnln1 x

dxd

b)1(

ln1

xb bx ln1


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2ln

1

)(log1. 2 xxdx

d

10ln

1

)(log2. xxdx

d


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30/09/2011 Techniques of differentiation 2130/09/2011Techniques of differentiation 21

Theorem

0everyfor

everyfor)(log

logxxa

Rxxa

x

x

a

a

1.

2.


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TheoremTheorem

If a > 1, the functionf(x) = logax is a

one-to-one, continuous, increasing function

with domain and range R. Ifx,y > 0

and ris any real number, then

),0(

yxxy baa loglog)(log.1

yxy

xbaa loglog)(log.2

xpx ap

a log)(log.3


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Hyberbolic Function


1. Hyperbolic sine ofx2

sinhxx ee

x

2. Hyperbolic cosine ofx 2cosh

xx eex

3. Hyperbolic tangent ofxxx

xx

ee

eextanh


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Graphs of some Hyperbolic

Functions


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Functions


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Functions


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Derivative of Hyperbolic Function


1. xxdx

dcosh)(sinh

xxdxd sinh)(cosh2.

xhx

dx

d 2sec)(tanh3.


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Continue


hxxhxdx

dsectanh)(sec4.

hxxhxdx

d csccoth)(csc5.

xhhx

dx

d 2csc)(coth6.


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1. xxdx

dcosh)(sinh

Proof:

)

2

()(sinhxx ee

dx

dx

dx

d

2

)1(xx ee

2

xx eexcosh


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Inverse Trigonometric

functions

The trigonometry functions are not

one-to-one.Thus they do not have inverse function


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To make the trigonometry function one-to-one, we restrict the domains ofthese functions.

Therefore, now, they have inversefunction.


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Illustration
http://en.wikipedia.org/wiki/Image:Sine_cosine_plot.svg


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The function22

,sin)( xxxf

is one-to-one.

Thus, the inverse of this restrictedsine function exists and is denoted

byxxf 11 sin)( xxf arcsin)(1or


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Example

Evaluate

)21(sin 1

)3

1

tan(arcsin

i.

ii.


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Properties of sine inverse

22-for,)(sinsin 1 xxx

11for,)sin(sin 1 xxx

i.

ii.

Cancellation Equation


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Example

Evaluate

)6.0sin(sin 1

)12

(sinsin 1

)3

2(sinsin 1

1.

2.

3.


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Derivative of Inverse Trigonometric

Function


2

1

1

1)(sin.1

xx

dx

d

2

1

1

1)(cos.2

xx

dx

d

2

1

1

1)(tan.3

xx

dx

d


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Continue


1

1)(csc.4

2

1

xxx

dx

d

1

1)(sec.5

2

1

xxx

dx

d

2

1

1

1)(cot.6

xx

dx

d


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Derivative of Inverse Hyperbolic

Function


2

1

2

1

2

1

1

1)(tanh.3

1

1)(cosh.2

1

1)(sinh.1

xx

dx

d

xx

dx

d

xx

dxd


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Continue


2

1

2

1

2

1

1

1)(coth.6

1

1)(sec.5

1

1)(csc.4

xxh

dx

d

xxxh

dx

d

xxxh

dx

d


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Derivative of functions

+

All the rules of differentiation such as

constant rule, sum rule, difference rule, product

rule and quotient rule.

+

Functions Identities


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4. The Sum Rule

Given functionsf(x) andg(x) which areboth differentiable, then

)]()([ xgxfdx

d

)()(

)()(

xgxf

xgdx

dxf

dx

d


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Proof)()()( xgxfxF

h

xFhxFxF

h

)()(lim)(

0

h

xgxfhxghxfxF

h

)]()([)]()([lim)(

0

h

xghxgxfhxfxF

h

)]()([)]()([lim)(

0

hxghxg

hxfhxfxF

hh

)]()([lim)]()([lim)(00

)()()( xgxfxF

Let


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6. The Product Rule

Given bothdifferentiable,then

)]()([ xgxfdx

d

)(and)( xgxf

)]([)( xgdx

dxf )]([)( xf

dx

dxg

or])()([ xgxf )()( xgxf )()( xfxg


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5. The Difference Rule

Given functionsf(x) andg(x) which areboth differentiable, then

)()(

)()()]()([

xgxf

xgdx

dxf

dx

dxgxf

dx

d


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Let F(x) =f(x)g(x)

then,

0

0

( ) ( )'( ) lim

( ) ( ) ( ) ( )lim

h

h

F x h F xF x

hf x h g x h f x g x

h


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In order to evaluate this limit, we would

like to separate the functionsf andgas

in the proof of the Sum Rule.

We can achieve this separation by subtracting

and adding the termf(x + h)g(x) in the numerator, as

follows.


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0

0

0 0 0 0

'( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )lim

( ) ( ) ( ) ( )lim ( ) ( )

( ) ( ) ( ) ( )lim ( ) lim lim ( ) lim

( ) '( ) ( ) '( )

h

h

h h h h

F x

f x h g x h f x h g x f x h g x f x g x

h

g x h g x f x h f xf x h g x

h h

g x h g x f x h f xf x h g x

h h

f x g x g x f x

7 Th Q ti t R l


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7. The Quotient Rule

Given thatfandgare differentiablefunctions,then

])(

)(

[ xg

xf

dx

d2)]([

)]([)()]([)(

xg

xg

dx

dxfxf

dx

dxg

[Order of differentiation is important]


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Proof

)(

)()(

xg

xfxQLet where Q,fandgare differentiable.


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)()()( xgxQxf

Now, rewrite as below, to use Product Rule

By the Product Rule

)()()()()( xgxQxgxQxf

Replace by)()()(

xgxfxQ


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)()(

)(

)()()( xgxg

xf

xgxQxf

)()()()()()( 2 xgxfxgxQxfxg

)()()()()()( 2 xgxfxfxgxgxQ

)(

)()()()()(

2

xg

xgxfxfxgxQ Proof


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Example 1

Find ify

x

xy

xxy

sin

4.2

.sin)4(1.3

3


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Example 2

Given ,1

1)(

t

ttf find ).1(f


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Solution

2)1)((

)1)(()(2

1)1)(()(

2

1

)( 21

21

21

21

21

t

tttt

tf


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2

)1)((

))]1()1(()[(2

1

)(21

21

21

21

t

ttttf

22 )1(

1

)1)((

)2()(2

1

)( 21

21

ttt

t

tf


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2)1(

1)(

tttf

2)11(1

1)1(f

4

1)1(f


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Example 3

cos1

cos)(

)(

f

f

if

Evaluate


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Solution

2)cos1(

)sin(cos)sin)(cos1()(f

2)cos1(

)cos1)((cos))(coscos1()(f

2)cos1(

cossincossinsin)(f


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Solution

2)cos1(

cossincossinsin)(f

2

)cos1(

sin)(f


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Example 4

Evaluate tttgg sec)()4

( 2if


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Solution

ttttttg tansec)2(sec)( 2

ttttttg tansecsec2)( 2

)4

tan()4

sec()4

()4

sec()4

(2)4

( 2g


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Solution

2)

4

(2)

2

()

4

( 2

g

)

8

1(

2

]

8

1[2)

2

()

4

(g


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Example 5

Compute the derivative of each of the

following;

tetg

f (x)

t

x

sin)(

4

ii.

i.

)tan21()(

1)( 3

ef

xexf

x

iv.

iii.


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Example

Find the derivative for each of the following:

xxy ln.1 1

ln

.2 2t

t

y

ty 2log

3

1.3 )(log.4 2

2 xxy


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Example 6

Evaluate the first derivative of each of the

following:

xxxf tanh)(.1 2

1sinh)(.2 f

sinh)(.3 f


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Example 7

xxy 12 tan)1(.1

1

tan.2

2

1

t

ty

Evaluate the derivative of each of the

following:


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derivative module 2 september 2011

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