Calcul de dérivées: solutions
1. dom f=R; dom f’=R f0(x) = 10¡x2− x + 2¢(2x − 1) 2. dom f=R; dom f’=R f0(x) = 2¡3x2− x + 3¢(6x − 1) 3. dom f=R∗; dom f’=R∗ f0(x) = 2x − 1 x2 = 2x3− 1 x2 4. dom f=R\ ½ 1 2 ¾ ; dom f’=R\ ½ 1 2 ¾ f0(x) = − 12 (2x − 1)3 5. dom f=R\ {5}; dom f’=R\ {5} f0(x) = 2 − 3 (x − 5)2 = 2x2− 20x + 47 (x − 5)2 6. dom f=R+; dom f’=R∗+ f0(x) = 2 + 1 2√x= 4√x + 1 2√x 7. dom f=R\©1 +√2, 1 −√2ª; dom f’=R\©1 +√2, 1 −√2ª f0(x) = − x 2− 7 + 8x (x2− 2x − 1)2 8. dom f= ·4 3, → ; dom f’= ¸4 3, → f0(x) = 2x − 1 2√x+ 3 2p(3x − 4) = 4x√xp(3x − 4) −p(3x − 4) + 3√x 2√xp(3x − 4) 9. dom f= ←, 2] \{−3}; dom f’= ←, 2[ \{−3} f0(x) = x − 7 2p(−x + 2) (x + 3)2 10. dom f= ¸ −12, → ; dom f’= ¸ −12, → f0(x) = 3x2+ 2x − 1 (2x + 1)p(2x + 1) 111. dom f=[1, → ; dom f’=]1, → f0(x) = 3x + 1 2p(x − 1) 12. domf=R∗ +;domf’=R∗+ f0(x) = − 1 2x√x 13. dom f =R; dom f’=R f0(x) =¡−2 + 28x + 9x2+ 26x3¢ ¡x3+ 2x − 1¢2(2x + 1)3 14. dom f =R\ ½ −14 ¾ ; dom f’=R\ ½ −14 ¾ f0(x) = 8x2+ 4x + 13 (4x + 1)2 15. domf=R\©1 +√6, 1 −√6ª;domf’=R\©1 +√6, 1 −√6ª f0(x) = 2(x2+ 16x − 11) (−x2+ 2x + 5)2 16. dom f =R\ ½ −12 ¾ ; dom f’=R\ ½ −12 ¾ f0(x) = 3 + 8 (2x + 1)3 = 24x3+ 36x2+ 18x + 11 (2x + 1)3 17. dom f =R\ {−1}; dom f’=R\ {−1} f0(x) = 4(x − 1) (x + 1)3 18. dom f =R; dom f’=R f0(x) = = (x + 1)2¡ 5x2− 2x + 2¢ 19. dom f = · 2 3, → ; dom f’= ¸ 2 3, → f0(x) = 2x − 2 + 3 2√3x − 2 20. dom f= ¸ 1 2, → ; dom f’= ¸ 1 2, → f0(x) = − 6x − 1 (2x + 1) (4x2− 1)√2x − 1 21. dom f =R+; dom f’=R∗+ f0(x) = 1 (√x + 1)2√x 2
22. dom f =[−2, → \ {1}; dom f’=]−2, → \ {1} f0(x) = − x2+ 10x + 13 2 (x − 1)3√x + 2 23. dom f=R∗ +; dom f’=R∗+ f0(x) = 3 µ √ x + 1 x ¶2 µ x2− 2√x 2x√x ¶ 24. dom f = ←, 1] ∪ · 3 2, → ; dom f’= ←, 1[ ∪ ¸ 3 2, → f0(x) = 4x − 5 2√2x2− 5x + 3 25. dom f =R; dom f’=R f0(x) = √ x 1 + x2 26. dom f =R\nq3 3 2 o ; dom f’=R\nq3 3 2 o f0(x) = 21x 2 (3 − 2x3)2 3
