Pravila diferenciranja

( )

( ) '''

''

)2

)1

vuvu

uCuC

+=+

=

( )

( )

( )

2

'''

'''

'''

)5

)4

)3

)2

v

vuvu

v

u

vuvuvu

vuvu

vuvu

⋅−⋅=

⋅+⋅=⋅

−=−

+=+

Tablica izvoda nekih elementarnih funkcija

( ) .,0'

constCC −=

( ) Rnnxxnn

∈=−

,

1'

( ) xx

ee =

'

( ) xx cossin'

=

( ) aaaxx

ln'

=

( )2

'

1

1arcsin

x

x

−

=( ) Rnnxx ∈= ,

( )x

x

xx 2

1,

11 '

2

'

=−=

( )ax

x

a

ln

1log

'

⋅

=

( ) { }0/,1

ln'

Rxx

x ∈=

( ) xx cossin =

( ) xx sincos'

−=

( )x

tgx2

'

cos

1=

( )x

ctgx2

'

sin

1−=

21 x−

( )2

'

1

1arccos

x

x

−

−=

( )2

'

1

1

xarctgx

+

=

( )2

'

1

1

xarcctgx

+

−=

Primjer 1. Odrediti izvode sledećih funkcija:

( ) ( )xxyb

xxy

xxxy

xxxy

xxxya

3213)

289

289

122433

243)

22

2'

012'

111213'

23

+⋅+=

+−=

+−=

⋅+⋅−⋅=

+−=

−−− ( ) 1'

−

=nn

nxx

( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

( ) ( )

xxy

xxxxxxxxy

xxxxxxy

xxxxy

xxxxy

xxxxy

vuvuvu

xxyb

2224

41812124121812

41433626

413326

0221332023

32133213

3213)

3'

3333'

22'

22'

122212'

'222

'2'

'''

+=

+++=+++=

⋅+⋅+⋅+⋅=

⋅+++⋅=

+⋅⋅+++⋅+⋅=

+⋅+++⋅+=

⋅+⋅=⋅

+⋅+=

−−

( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( ) ( ) ( )( )( )

( ) ( )

2

''''

'

2

'

'

1ln11

ln1

1

1ln1lnln

1

1ln1ln

1

ln)

'

⋅−+⋅

⋅+⋅

+

+⋅−+⋅⋅+⋅=

+

+⋅−+⋅=

+=

xxxxx

x

xxxxxxxxy

x

xxxxxxy

x

xxyc

2

'''

v

vuvu

v

u ⋅−⋅=

( ) ( )

( )( ) ( )

( )

( )

( )2'

2

'

2

'

2

'

1

1ln

1

ln1lnln

1

ln11ln

1

1ln1ln1

+

++=

+

−+++=

+

−+⋅+=

+

⋅−+⋅

⋅+⋅

=

x

xxy

x

xxxxxxy

x

xxxxy

x

xxxx

xx

y

Zadaci za vježbu: Odrediti izvode sledećih funkcija

( ) ( )xxyd

xxyc

xxyb

xxxya

cos)

67)

46)

8435)

2

22

43

56

⋅=

+⋅−=

−=

−+−= Uputstvo:Koristiti tablicu

izvoda i osnovna pravila izvoda.

( ) xxxxyh

xxxyg

x

xyf

x

xxye

sin2cos2)

cossin)

ln)

1

1)

2

2

2

2

⋅+⋅−=

+⋅=

=

+

+−=