Reguli+Formule de Derivare

Reguli de derivare

1.

(α ⋅ f )′ = α ⋅ f ′

;

' 2. ( f ± g ) = f ' ± g ' ;

 n  ' n   3. ∑ f = ∑ f ' ;  k  k  k =1  k =1 4. ( f ⋅ g )

'

= f ' g + g ' f ;

'   n n    5. ∏ f = ∑  f ⋅ f ⋅K⋅ f ' ⋅K⋅ f n   ;   k 1 2 k    k =1  k =1 '  f  f ' g − g ' f 6.   ;  = 2 g   g '  1  − g ' 7.    = 2 ; g   g ' 8. ( f (u )) = f ' (u ) ⋅ u' ;

( f (u(v)))' = f ' (u(v ))⋅ u' (v)⋅ v'

9. 10. 10 .

y

.

' ' 1 1  f −1  =   − 1 sau  f   , unde    y0   =   f ' x     f '    − 1  0   f     

= f  x    . 0  0 

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-2-

Tabel de derivare al fun functiilor ctiilor elementare Nr. Nr.

Functia

Derivata

Domeniul de derivabilitate

1 2 3

c (constanta) x

0 1 x

x ∈ R x ∈ R x ≠ 0

x

x 4

x n , n ∈ N ∗

nx n−1

x ∈ R

5

x r , r ∈ R

rx r −1

x ∈ (0 , ∞ )

x

1

x ∈ (0 , ∞ )

x , n ∈ N ∗

2 x 1

6 7

n

ln x

nn x n −1 1

ln x

x 1

log log a x

x 1

e x

x ln a e x

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

a x , a

> 0, a ≠ 1

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

x ∈ (0 , ∞ ) x ∈ R x ∈ R

a x ln a cos x − sin x

sin x cos x tg x

1

ctg x

cos 2 x 1

−

= 1 + tg 2 x

= −(1 + ctg 2 x )

sin sin 2 x

≠0

x ∈ R x ∈ R x

≠

π

x

≠ k π , k ∈ Z

2

+ k π , k ∈ Z

arcsin x

1

x ∈ (− 1,1)

arccos x

1 − x 2 1

x ∈ (− 1,1)

−

1 − x 2 1

arctg x arcctg x

ch x =

e x

+ e − x

2 e x − e − x

1 + x 2 1

−

x ∈ R

1 + x 2

sh x =

e x

− e − x 2

e x

x ∈ R

+ e − x

x ∈ R x ∈ R



-3-

Tabel de derivare derivar e al functiilor functiilor compuse Nr. Nr.

Functia

Derivata

1

u n , n ∈ N∗

n ⋅ u n−1 ⋅ u '

2

u r , r ∈ R

r ⋅ u r −1 ⋅ u '

3 4

n

u'

u

2 u u'

ln u

n ⋅ n u n−1 u'

u

≠0

log log a u

u u'

u

>0

9

au , 0, a

12

>

u ⋅ ln a eu ⋅ u '

u

a u ⋅ u ' ⋅ ln a

≠1

cos u ⋅ u '

sin u cos u tg u

− sin u ⋅ u ' 1 2

13 14 15

n par u ≠ 0 pentru n impar

ln u

e

11

> 0 pentru

u u'

8

10

u

≠0

6

a

>0 u >0 u

u

5

7

u

Coditii de derivabilitate derivabilit ate

ctg u

−

cos u 1 2

sin sin u

⋅ u ' = (1 + tg 2u )⋅ u ' ⋅ u ' = −(1 + ctg2u )⋅ u '

cos u ≠ 0 sin sin u ≠ 0

arcsin u

u'

u2

<1

arccos u

1−u2 u'

u2

<1

−

16

arctg u

17

arcctg u

18

sh u

19

ch u

1−u2 1 ⋅u' 1+u2 1 − ⋅u' 1+u2 ch u ⋅ u ' sh u ⋅ u '

20

uv

u v ⋅ ln u ⋅ v '+ v ⋅ u v −1 ⋅ u '

Reguli+Formule de Derivare

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