Reguli+Formule de Derivare [PDF]

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Reguli de derivare ′ 1. (α ⋅ f ) = α ⋅ f ′ ; ' 2. ( f ± g ) = f ' ± g ' ; 3.

 n    f ∑   k  k =1 

'

n = ∑ f '; k =1 k

' 4. ( f ⋅ g ) = f ' g + g ' f ; 5.

 n    f  ∏ k  k =1 

'

n = ∑  f ⋅ f ⋅K⋅ f ' ⋅K⋅ f n  ; k  k =1 1 2

' f  f ' g − g ' f 6. = ; g  g2    

7.

1   g  

'

=

−g' ; g2

' 8. ( f (u )) = f ' (u ) ⋅ u' ; 9. 10.

( f (u(v)))' = f ' (u(v ))⋅ u' (v)⋅ v' .   

'  − 1 f  = 

1

f '  f −1  

y = f  x  . 0  0



'  − 1 sau f   y   

  0   

=

1 , unde   f ' x   0

-2-

Tabel de derivare al functiilor elementare Nr. 1 2 3

Functia c (constanta) x x

Derivata 0 1 x x

Domeniul de derivabilitate x∈R x∈R x≠0

4 5 6

xn , n ∈ N∗ xr , r ∈ R

nx n−1 rx r −1 1

x∈R x ∈ (0 , ∞ ) x ∈ (0 , ∞ )

7

x n

x , n∈N

2 x 1

∗

n

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

n −1

n x 1 x 1 x 1 x ln a ex a x ln a cos x − sin x

x ∈ (0 , ∞ )

8

ln x

9

ln x

10

log a x

11 12 13 14 15

ex a x , a > 0, a ≠ 1 sin x cos x tg x

16

ctg x

17

arcsin x

1 = 1 + tg 2 x cos 2 x 1 − = − 1 + ctg 2 x 2 sin x 1

arccos x

1− x 1

18 19

arctg x

20

arcctg x

21

e x + e −x ch x = 2 x e − e −x sh x = 2

22

x≠0 x ∈ (0 , ∞ )

x∈R x∈R x∈R x∈R

(

−

π + kπ , k ∈ Z 2 x ≠ kπ , k ∈ Z

x≠

)

x ∈ (− 1,1)

2

x ∈ (− 1,1)

1− x 1 1 + x2 1 − 1 + x2 e x − e −x sh x = 2 x e + e −x ch x = 2 2

x∈R x∈R x∈R x∈R

-3-

Tabel de derivare al functiilor compuse Nr. 1 2 3 4

Functia u , n∈N n

Derivata ∗

u , r ∈R r

n ⋅u

n −1

⋅u'

r ⋅u

r −1

⋅u'

u >0 u >0

u' 2 u u'

u n

Coditii de derivabilitate

u

u > 0 pentru n par u ≠ 0 pentru n impar

n ⋅ n u n−1 u' u u' u u' u ⋅ ln a eu ⋅ u '

u≠0

5

ln u

6

ln u

7

log a u

8

eu

9 10 11 12

au , a > 0, a ≠ 1 sin u cos u tg u

13

ctg u

14

arcsin u

1 ⋅ u ' = 1 + tg 2u ⋅ u ' 2 cos u 1 − ⋅ u ' = − 1 + ctg 2u ⋅ u ' 2 sin u u'

arccos u

1−u2 u'

15

u≠0 u >0

a u ⋅ u '⋅ ln a cos u ⋅ u ' − sin u ⋅ u '

(

)

(

−

)

16

arctg u

17

arcctg u

18 19 20

sh u ch u

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

uv

u v ⋅ ln u ⋅ v '+ v ⋅ u v −1 ⋅ u '

cos u ≠ 0 sin u ≠ 0 u2