Derivatives Module 1 September 2011

8/12/2019 Derivatives Module 1 September 2011

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26/09/2011Derivatives - by definition 2

Learning Objective

At the end of the module, studentsshould be able to

compute derivative of a functionat arbitrary point using definition.

determine whether a functionis differentiable or not.


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Secant line

x0 x0+ hx

y

P f ( x0 +h)

f ( x0)

0

y = f ( x)Q

h x f h x f

xh x x f h x f

m PQ)()(

)()()( 00

00

00

Slope of the secant line PQ:


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x0 x0+ hx

y

P f ( x0 + h)

f ( x0)

0

y = f ( x)

Q

, provided the limit exists.

Derivative as slope to a curve y = f ( x)at the point x = xo.

which can be represented as limit below:

h x f h x f

mh

)()(lim 00

0

By moving the red dot closer andcloser to the black dot we canapproximate slope on the curve at point P.


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Derivative of a function

The derivative (slope) of a function f atwith respect to the variable x,is

is defined by

),( x f

h

x f h x f

x f h)()(

lim)( 0


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7/4726/09/2011Derivatives - by definition 7

Example 1 - solution ,)( 2 x x x f

h x f h x f x f

h)()(lim)(

0

(i)



Example 1 - solution ,)( 2 x x x f

h

x xh xh x

h

)][()]()[(lim

22

0

h

x xh xh xh x

h

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

222

0

(i)



Example 1 - solution

h

hh xh x f

h

2

0

2lim)(

1212lim)(0

xh x x f h

(i)

hh xh

x f h

)12(lim)(

0



Example 1 - solution 12)( x x f

(ii) ,)( 2 x x x f

-1/41/2 1

x

y

0

Domain?



Example 2

Find the derivative of ,1)( x x f

Is there any difference between the domain

of ? and f f



Example 2- solution ,1)( x x f

h x f h x f

x f h

)()(lim)(

0

h xh x

h

11)(lim

0




11)(

11)(.

11)(lim)(

0

xh x

xh x

h

xh x x f

h

11)(

)1(1)(lim

0

xh xh

xh xh




)11)((lim)(

0 xh xh

h x f

h

12

1

11)(

1lim

0

x xh xh


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12

1)(

x x f ,1)( x x f

Domain Domain


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Example 3

Find .22

)(if )(t t

t g t g

What is the difference between thedomain of ?and g g


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Example 3 - solution .22

)(if )(t t

t g t g

h

t g ht g t g

h

)()(lim)(

0

ht t

ht ht

t g h

]22

[])(2)(2

[

lim)( 0


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)2)(2()2224()2224(

lim)(22

0 t ht hht t t ht ht ht t t

t g h

)2)(2(4

lim)(0 t ht h

ht g

h


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2

0

)2(4

)(

)2)(2(

4lim)(

t t g

t ht t g

h


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2)2(4

)(t

t g

2exceptnumberrealalldomain22

)( t t t t

t g

2exceptnumberrealalldomain t t

No difference in their domain.


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Example 4

Find the derivative of the function

,13)(2

x x x f

at the number x = 2.


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Example 4 - solution ,13)( 2 x x x f

By

h x f h x f x f

h)()(lim)( 00

00


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h f h f

f h

)2()2(lim)2(

0

hhh

f h

]12)2(3[]1)2()2(3[lim)2(

22

0


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hhh

f h]12)2(3[]1)2()2(3[

lim)2(

22

0

h

hhh f

h

]11[]1)2()44(3[lim)2(

2

0

11113lim113

lim)2(0

2

0h

hhh

f hh


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Example 5

Given that f ( x) = x3 + 1

(a) Find the average rate of change of f with respect to x over the interval[ 2, 6].

(b) Find the instantaneous rate ofchange of f with respect to x at

x = - 2.


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(a)

4)12()16(

26)2()6( 33 f f

r ave

524

20826

9217

h x f h x f )()(

changeof rateaverage 00

f ( x) = x3 + 1


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h f h f

h x f h x f

r

h

hinst

)2()2(lim

)()(lim

0

00

0

(b)

h

hh

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

lim

33

0

f ( x) = x3 + 1


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(b)

hhhh

h

)2)2)(2()2)((22(lim

22

0

hhhhh

h

)42444)((lim

2

0

121200)126(lim2

0 hhh


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PracticeFind the derivative of the given function usingdefinition and state the domain of the functionthe domain of its derivative.

1)( x x f (i)

2

1)( x x f (ii)


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Some other notations for derivative

If y = f ( x), then the derivative can also be

denoted as

dxdf

dxdy

y x f ),(

)()()( x f D x Df x f dxd

x


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Definition

A function f is differentiable at c ifexists.

)(c f


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Definition

It is differentiable on an open interval ( a ,b)

[ or or or ] if it is

differentiable at EVERY NUMBER IN THEINTERVAL .

),(a ),( a ),(


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Example

Determine, where the function

x x f )( is differentiable?


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Example

Recall

0,0,)(

x x x x x x f

0 x

y


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Example - solution

Case 1: If x > 0

h x f h x f x f

h)()(lim)(

0

11lim)(

lim00 hh

h

xh x

Therefore, f ( x) is differentiable for all x when x > 0 .

0,

0,)(

x x

x x x x f


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Example - solution

Case 2: If x < 0

h x f h x f x f

h)()(lim)(

0

1)1(lim)()(

lim00 hh

h

xh x

Therefore, f ( x) is differentiable for all x when x < 0 .

0,

0,)(

x x

x x x x f


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Example - solution

x

y

0

x x f )( x x f )(

Case 3: If x = 0

h x f h x f x f

h)()(lim)(

0

h

hh

00lim0 does not exists.


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Example - solution

x

y

0

x x f )( x x f )(

Case 3: If x = 0

100

lim0 h

hh

100

lim0 h

hh

and

Therefore, is not differentiable at x = 0 . x x f )(

The domain where is differentiable is x x f )(),0()0,(


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Example - solution

x

y

0

x x f )( x x f )(

The domain where is differentiable is x x f )(

),0()0,(


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Theorem

If f is differentiable at x = a , then f iscontinuous at x = a .

Note that the opposite (or converse) of the theoremis not always true (See previous example).


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3 possibilities a function is notdifferentiable.

y

x0

If the graph of the function

has a sharp corner init, the the graph has notangent at this point and thereforethe function is not differentiable at that point.

1.


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y

x0

If the graph of the function

has jump discontinuity (i.e thefunction discontinuous), thenthe function is not differentiable.

2.


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y

x0

If the graph of the function f has vertical tangent at x = a ,that is, f is continuous at x = a ,however

3. (Infinite discontinuities)

.)(lim x f a x

a

This means that the tangent lines becomes steeper and steeper as .a x


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PracticeShow that

1,2

1,2)(

2

x x

x x x f

is continuous but not differentiable at x = 1 .Sketch the graph of f and . f


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END OF THE MODULE 1

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Derivatives Module 1 September 2011