Réponse:
a. f(x)' = -6x² + 14x + 3/9
b. f(x)' = 16x - 35/49
c. f(x)' = 6x² + 3x² - 3
d. f(x)' = -12x² + 12x - 6
f. f(x)' = 2/x³
g. f(x)' = -2x² + 2x + 12
Explications étape par étape:
a. • (-2x³)' = -2×(3x²) => -6x²
• (7x²)' = 7×2x => 14x
• (x/3)' => (u/v)' = (u×v' - v×u)/v²
u = x ; u'= 1 ; v = 3 ; v' = 0
(x/3)' = (x × 0 - 3 × 1)/3² = -3/9
• -6x² + 14x + 3/9
b. • (8x²)' = 8×2x => 16x
• ((5x - 2)/7)' = (u/v)' = (u×v' - v×u)/v²
u = 5x - 2 ; u' = 5 ; v = 7 ; v' = 0
• ((5x - 2)/7)' = (5x × 0 - 7 × 5)/7² = -35/49
• 16x - 35/49
c. • (3x (x² - 1))' = (u×v)' => u × v' + v × u
u = 3x ; u' = 3 ; v = x² - 1 ; v' = 2x
(3x (x² - 1))' = 3x × 2x + (x² - 1)×3
= 6x² + 3x² - 3
d. • ((-2x + 3) (2x² + 3))' = (u×v)' => u × v' + v × u
u = -2x + 3 ; u' = -2 ; v = 2x² + 3 ; v' = 2×2x = 4x
((-2x + 3) (2x² + 3))' = (-2x +3) × 4x + (2x² + 3)× (-2)
= -8x² + 12x -4x² - 6
= -12x² + 12x - 6
e. • (5racine carré de x/(3x + 1))' = (u×v)' => u × v' + v × u
f. • (1/x²)' = (u/v)' = (u×v' - v×u)/v²
u = 1 ; u' = 0 ; v = x² ; v' = 2x
(1/x²)' = (1 × 2x - x² × 0)/(x²)²
= 2x/x⁴ => 2/x³
g. • ((-2x + 1)/(x² - 6x + 9))' = (u/v)' = (u×v' - v×u)/v²
u = -2x + 1 ; u' = -2 ; v = x² - 6x + 9 ; v' = 2x - 6
((-2x + 1)/(x² - 6x + 9))' = ((-2x + 1) × (2x - 6)) - ((x² - 6x + 9) × (-2))/ (x² - 6x + 9)²
= -4x² + 12x + 2x - 6 + 2x² - 12x + 18
= -2x² + 2x + 12
