Reguli de derivare

1. ( ) ff ′⋅=′⋅ αα ;

2. ( ) ''' gfgf ±=± ;

3. ∑=

=∑=

n

k kfn

k kf 1'

'

1 ;

4. ( ) fggfgf ''' +=⋅ ;

5. ∑=

⋅⋅⋅⋅⋅=∏=

n

knfkfff

n

k kf 1'

21

'

1KK ;

6. 2

'''

gfggf

gf −=

;

7. 2

''1g

gg

−=

;

8. ( )( ) ( ) ''' uufuf ⋅= ;

9. ( )( )( ) ( )( ) ( ) '''' vvuvufvuf ⋅⋅= .

10.

−=−

1'

1'1ff

f sau

=−

0'1

0'1

xfyf , unde

= 00 xfy .

- 2 -

Tabel de derivare al functiilor elementare Nr. Functia Derivata Domeniul de derivabilitate 1 c (constanta) 0 R∈x 2 x 1 R∈x 3 x

xx

0≠x

4 nx , ∗∈ Nn 1−nnx R∈x 5 rx , R∈r 1−rrx ( )∞∈ ,0x 6 x

x21

( )∞∈ ,0x

7 n x , ∗∈ Nn n nxn 1

1−

( )∞∈ ,0x daca n este par 0≠x daca n este impar

8 xln x1

( )∞∈ ,0x

9 xln x1

0≠x

10 xalog ax ln

1

( )∞∈ ,0x

11 xe xe R∈x 12 xa , 1,0 ≠> aa aa x ln R∈x 13 xsin xcos R∈x 14 xcos xsin− R∈x 15 xtg

xx

22

tg1cos

1+= π

πkx +≠

2, Z∈k

16 xctg ( )xx

22

ctg1sin

1+−=−

πkx ≠ , Z∈k

17 xarcsin 21

1

x−

( )1,1−∈x

18 xarccos 21

1

x−−

( )1,1−∈x

19 xarctg 21

1x+

R∈x

20 xarcctg 21

1x+

− R∈x

21

2ch

xx eex

−+=

2sh

xx eex

−−=

R∈x

22

2sh

xx eex

−−=

2ch

xx eex

−+=

R∈x

- 3 -

Tabel de derivare al functiilor compuse

Nr. Functia Derivata Coditii de derivabilitate 1 nu , ∗∈ Nn '1 uun n ⋅⋅ −

2 ru , R∈r '1 uur r ⋅⋅ − 0>u 3 u

uu

2'

0>u

4 n u n nun

u1

'−⋅

0>u pentru n par

0≠u pentru n impar

5 uln uu '

0≠u

6 uln uu '

0≠u

7 ualog au

uln'

⋅

0>u

8 ue 'ueu ⋅

9 ua , 1,0 ≠> aa

auau ln'⋅⋅

10 usin 'cos uu ⋅ 11 ucos 'sin uu ⋅− 12 utg ( ) 'tg1'

cos1 2

2 uuuu

⋅+=⋅ 0cos ≠u

13 uctg ( ) 'ctg1'sin

1 22 uuu

u⋅+−=⋅−

0sin ≠u

14 uarcsin 21

'

u

u

− 12 <u

15 uarccos 21

'

u

u

−− 12 <u

16 uarctg '

11

2u

u⋅

+

17 uarcctg '

11

2u

u⋅

+−

18 ush 'ch uu ⋅ 19 uch 'sh uu ⋅ 20 vu ''ln 1 uuvvuu vv ⋅⋅+⋅⋅ −