MFB-Differentation-1

7/26/2019 MFB-Differentation-1

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Differentiation


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Differentiation Using Limits of

Difference Quotients

DEFINITION:

The slope of the tangent

lineat (x, f(x)) is

This limit is also the

instantaneous rate of

changeoff(x) atx.

m=limh0

f x+h( ) f x( )h

.


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DEFINITION:

For a functiony = f (x),

its derivativeatxis the

function defined by

provided the limit exists.

If exists, then we say

that f is differentiableatx.

f x( )=limh0

f x+h( ) f x( )h

f x( )


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Eample: For find . Then

find and .

f x( )f x( )= x2 ,

f 3( ) f 4( )f x( )=lim

h0

f x+h( ) f x( )h

=limh0

x+h( )2 x2h

f x( )=limh0

x2 +2xh+h2 x2

h=lim

h0

h 2x+h( )h

f x( )=limh0

2x+h

f x( )=2x

f 3( )=2 3( )= 6 f 4( )=2 4( )=8


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Eample : For find .

Then find and .

f x( )= x3, f x( )f 1

( ) f 1.5

( )f x( )=lim

h0

f x+h( ) f x( )h

=limh0

x+h( )3 x3h

f x( )=limh0 x3

+3x2

h+3xh2

+h3

x3

h

f x( )=limh0

h 3x2 +3xh+h2( )h

=limh0

3x2 +3xh+h2

f x( )=3x2


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!here a Function is Not Differentiable:

1) A functionf(x) is not differentiable at a pointx = a,if there is a corner! at a.

") A function f (x) is not

differentiable at a point

x = a, if there is a vertical

tan#ent at a.


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Leibni"#s Notation:

$henyis a function ofx, we will also desi#nate thederivative, , as

which is read the derivative ofywith respect tox.!

f x( )

Differentiation Techni$ues:

The %o&er 'ule and (um)Difference 'ules

dy

dx,


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T*EO'E+ ,: The %o&er 'ule

For any real number k, dy

dxxk = kxk1

Eample ,: %ifferentiate each of the followin#&

a) b) c)

a) b) c)d

dxx = 1x11

d

dxx = 1

d

dxx

5

= 5 x51

d

dxx

5 = 5x4

d

dx

x4 = 4x41

d

dxx4 = 4x5

y= x4y=xy=x5


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Eample : %ifferentiate&

a) b)

dy

dxx =

dy

dxx

1

2 = 1

2x

1

21

dy

dxx =

1

2x

12 , or

= 1

2x1

2

, or

= 1

2 x

d

dx

x0.7 = 0.7 x0.71

d

dxx

0.7 = 0.7x0.3

y=x0.7y= x


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T*EO'E+ -:

The derivative of a constant times a function is the

constant times the derivative of the function. That is,

d

dxc f(x)[ ] = c d

dxf(x)

T*EO'E+ .:

The derivative of a constant function is '. That is,d

dxc = 0


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Eample : Find each of the followin# derivatives&

a) b) c)

d

dx

7x4 d

dx

(9x)d

dx

1

5x

2

d

dx7x4 = 7

d

dxx

4

= 7 4x41

d

dx7x4 = 28x3

d

dx(9x) = 9

d

dxx

= 9 1x11

ddx

(9x) = 9d

dx

1

5x2

= 1

5

d

dx

1

x2

= 1

5

d

dxx

2

= 15

2x21

d

dx

1

5x2

= 2

5x

3, or= 2

5x3


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T*EO'E+ /: The (um)Difference 'ule

Sum& The derivative of a sum is the sum of the

derivatives.

Difference& The derivative of a difference is the

difference of the derivatives.

d

dx f(x) +g(x)[ ] = d

dx f(x) + d

dx g(x)

d

dxf(x) g(x)[ ] = d

dxf(x)

d

dxg(x)


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a) ddx

(5x3 7)

d

dx(5x3 7) =

d

dx(5x3 )

d

dx(7)

= 5 d

dxx

3 0= 5 3x31

d

dx(5x3 7) = 15x2


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ddx

24x x + 5x

b)

d

dx24x

x

+

5

x

=

d

dx(24x)

d

dxx

+

d

dx

5

x

=24 d

dxx

d

dxx

1

2 + 5 d

dxx

1

=24 1x11 1

2x

1

2

1

5 1x11

=241

2x

1

2 +5x2 , or =24 1

2 x+

5

x2



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Differentiation Techni$ues:

The %roduct and Quotient 'ules

T*EO'E+ 0: The %roduct 'ule

et Then,F(x) = f(x) g(x).

F(x) = d

dxf(x) g(x)[ ]

F(x) = f(x) ddx

g(x) + g(x) ddx

f(x)


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Eample : Find ddx

x4 2x3 7( ) 3x2 5x( ) .

d

dxx4 2x3 7( ) 3x

2 5x( ) =x

4 2x3 7( ) 6x5( )+ 3x2 5x( ) 4x3 6x2( )


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T*EO'E+ 1: The Quotient 'ule

If then,Q(x)= N(x)

D(x)

,

Q(x) = D(x) N(x) N(x) D(x)

D(x)[ ]2


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Eample : %ifferentiate f(x) = x2 3xx1

.

f (x) = (x

1)(2x

3)

(x2

3x)(1)

(x1)2

f (x) = 2x2 5x+3x2 +3x

(x1)2

f (x) = x2 2x+3

(x1)2


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The 2hain 'ule

T*EO'E+ 3: The Etended %o&er 'ule

Suppose that g(x) is a differentiable function

of x. Then, for any real number k,

d

dx

g x( ) k

=k g x( ) k1

d

dx

g x( )


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Eample : %ifferentiate f x( )= 1+x3( )1

2 .

ddx

1+x3( )1

2 =12 1+x3

( )1

21

3x2

=3x2

21+x3( )

1

2

= 3x2

2 1+x3


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Eample :

%ifferentiate

ombine *roduct +ule and xtended *ower +ule

-implified&

f x( )= 3x5( )4 7 x( )10 .

f x( )= 3x5( )4 10 7 x( )9 1( )+4 3x5

( )

37 x

( )

103

( )

f x( )=2 3x5( )3 7 x( )9 67 21x( )


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Eample : For and

find and

dy

dx=

dy

dudu

dx

= 1

2 u 3x2 =

3x2

2 x3 +1

y=2 + u u= x3 +1,

dydu

, dudx

, dydx

.

dy

du=

1

2u

1

2du

dx

=3x2

and


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Eample : For and

find

dy

dt=

dy

dudu

dt= (2u3)(5)

= 10u15= 10(5t1) 15

= 50t10 15= 50t25

y = u2 3u u = 5t1,dy

dt .

dy

du= 2u3

du

dt= 5and


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T*EO'E+ 4

ddx

ex =ex


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Eample : Find dydx&

a) y=3ex; b) y=x2ex; c) y= ex

x3.

a)dy

dx

3ex

( ) = 3

d

dx

ex

= 3ex

b) ddx

x2ex( ) = x2 ex +ex 2x

= ex x2 +2x( )


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Eample 5concluded6:

c)

d

dx

ex

x3

=

x3 ex ex 3x2

x3( )2

= x

2ex x3( )

x6

= e

x (x3)x

4


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T*EO'E+ 7

or

The derivative of eto some power is the product of e

to that power and the derivative of the power.

d

dxe

f(x ) =ef(x ) f (x)

d

dxeu =eu

du

dx


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Eample : %ifferentiate each of the followin# with

respect tox&

a) y=e8x ; b) y=ex2+4x7; c) y=e x

23 .

a) ddx

e8x = e8x 8

= 8e8x

b)d

dxe

x2 +4x7 = ex2 +4x7 2x+4( )


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Eample 5concluded6:

c)d

dx e x23

= d

dx e x

23

( )

1

2

= e x23( )

1

2

1

2x

2 3( )1

2 2x

= xe

x23

x2 3


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Logarithmic Functions

T*EO'E+ ,8

For any positive numberx,

d

dxlnx=

1

x.


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T*EO'E+ ,,

or

The derivative of the natural lo#arithm of a function is

derivative of the function divided by the function.

d

dxlnf(x) =

1

f(x) f (x)=

f (x)f(x)

,

d

dxlnu=

1

udu

dx.


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Eample 4: %ifferentiate

a)y=ln(3x);

b)y=ln(x2 5); c)f(x)=ln(lnx); d) f(x)=ln x3 +4x

.

a)dy

dx

= 1

3x

3 =1

x

.

b)dy

dx=

1

x2 5

2x= 2x

x2 5

.

c) f (x)= 1lnx

1x

= 1x lnx

.


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Eample 4 5concluded6:

d) f (x) =

d

dx [ln(x

3

+4) lnx]

= 3x2

x3 +4

1

x

= 3x2

x1 (x3

+4)x(x3 +4)

= 3x

3 x3 4

x(x3

+4)

= 2x3 4

x(x3 +4)


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Derivatives of Trigonometric Functions


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36/44


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*igher Order Derivatives

*igher)Order Derivatives:

onsider the function #iven by

Its derivative f is #iven by

The derivative function f can also be differentiated.$e can thin/ of the derivative f as the rate of chan#eof the slope of the tan#ent lines of f . It can also be

re#arded as the rate at which is chan#in#.

y = f(x) = x5 3x4 +x.

y = f (x) = 5x4 12x3 +1.

f x( )


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*igher)Order Derivatives 5continued6:

$e use the notation f for the derivative .That is,

$e call f the second derivative of f. For

the second derivative is #iven by

f (x) = d

dxf (x)

y = f(x) = x5 3x4 +x,

y = f (x) = 20x3 36x2 .

f( )


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ontinuin# in this manner, we have

$hen notation li/e #ets len#thy, we abbreviate

it usin# a symbol in parentheses. Thus is thenthderivative.

f (x) = 60x2 72x, the third derivative of f

f (x)= 120x72, the fourth derivative of f

f (x)= 120, the fifth derivative of f.

f x( )f n( ) x( )


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For we havey = f(x) = x5 3x4 +x,

f( 3) (x) = 60x2 72x,

f( 4 )(x) = 120x72,f

( 5)(x) = 120,

f( 6)(x) = 0, and

f( n)(x) = 0, for any integer n6.


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eibni0s notation for the second derivative of anotation for the second derivative of a

function #iven by y 2 f(x) isfunction #iven by y 2 f(x) is

read the second derivative of y with respect to x.!read the second derivative of y with respect to x.!

The "s in this notation are 34T exponents.The "s in this notation are 34T exponents.

d2y

dx2 , or d

dx

dy

dx


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*igher)Order Derivatives 5concluded6:

If theny = x5 3x4 +x,

dy

dx= 5x4 12x3 +1,

d4y

dx4 = 120x72,

d2y

dx2

= 20x3 36x2 , d

5y

dx5

= 120.

d3ydx

3 = 60x2 72x,


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Eample : For find

y = x1

dy

dx= x2

d2y

dx2 = 2x

3, or 2

x3

d2y

dx2.y =

1

x,


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