Clasa XI – lic rută directă / XII – lic seral frecv redusă / XII – lic rut ă progresivă – prof. Cornelia Mestecan Fişă de lucru 1 – reguli de derivare, derivatele funcţiilor simple – formule REGULI DE DERIVARE

1. ( )' ' 'f g f g+ = +

2. ( )' 'c f c f⋅ = ⋅

3.( )' ' 'f g f g f g⋅ = ⋅ + ⋅

4.' ' '

2

f f g f g

g g

⋅ − ⋅=

5. ( ) ( )' ' 'f u f u u= ⋅�

Funcţia Derivata Mul ţimea pe care funcţia este derivabilă

Funcţia compusă Derivata

C (constantă) 0 ℝ - - x 1 ℝ u 'u

nx 1nn x −⋅ ℝ nu 1 'nn u u−⋅ ⋅ rx 1rr x −⋅ ( )0;+∞ ru 1 'rr u u−⋅ ⋅

x 1

2 x ( )0;+∞ u '1

2u

u

ln x 1

x ( )0;+∞ ln ( 0)u u > '1

uu

xe xe ℝ ue 'ue u⋅ ( 0; 1)xa a a> ≠ lnxa a⋅ ℝ ( 0; 1)ua a a> ≠ 'lnua a u⋅ ⋅

sinx cosx ℝ sinu '(cos )u u⋅ cosx sinx− ℝ cosu '( sin )u u− ⋅ tgx

2

1

cos x /

2k k

π π − + ∈

ℝ ℤ tgu

'2

1

cosu

u⋅

ctgx 2

1

sin x− { }/k kπ− ∈ℝ ℤ ctgu

'2

1

sinu

u− ⋅

arcsinx 2

1

1 x− ( )1;1− arcsinu '

2

1

1u

u⋅

−

arccosx 2

1

1 x−

− ( )1;1− arccosu '

2

1

1u

u− ⋅

−

arctgx 2

1

1 x+

ℝ arctgu '

2

1

1u

u⋅

+

arcctgx 2

1

1 x−

+

ℝ arcctgu '

2

1

1u

u− ⋅

+