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formule de derivare
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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 ⋅⋅+⋅⋅ −