x2

′
(a) = 0
′
(ax) = a
2 ′
x = 2x
(xα )′ = αxα−1
√
1
( x) = √
′ 2 x
1
= −1
x2
x
′
1
−2
= 3
2
x
x
′
1
−3
= 4
x3
x
POCHODNE - WZORY
′
(sin x) = cos x
′
′
(ln x) = x1 ,
′
(sin ax) = a cos ax (ex ) = ex ,
(ln ax) =
a
ax
(eax ) = aeax
′
(f (x) ± g (x)) = f ′ (x) ± g′ (x)
′
(af (x)) = af ′ (x)
1
′
(arcsin x) = √
(f (x) · g (x)) = f ′ (x) g (x) + f (x) g′ (x)
1 − x2
′
3 ′
1
1
f (x)
f ′ (x) g (x) − f (x) g ′ (x)
′
′
2
(tg x) =
(arctg
x)
=
=
x = 3x
cos2 x
1 + x2
g (x)
(g (x))2
4 ′
−1
−1
′
′
′
x = 4x3
(ctg x) =
[f (g (x))] = f ′ (g (x)) · g′ (x)
(arccos x) = √
2
2
sin
x
1
−
x
_________________________________________________________________________
′
(cos x) = − sin x
POCHODNE - ZADANIA
Pochodne sum i iloczynów przez liczbȩ:
√
2
x+2 2+ 2
+
;
1. f (x) = 2x2 − 3x3 + ; 2. f (x) =
x
2
x
√
√
√
x 2
1
3. f (x) = (2x)2 + 2x; 4. f (x) = 3x2 + − + 2; 5. f (x) = x x + √ ; 6. f (x) = 2 sin x − 3 cos x;
2 x
x x
7. f (x) = 2 ln x + x ln 2; 8. f (x) = arcsin x + arccos x; 9. f (x) = arctg x + arcctg x; 10. f (x) = tg x − ctg x.
Pochodne iloczynów i ilorazów
ln x
x+3
; 16. f (x) =
;
x
3−x
2
x + ln x
21 f (x) = 2
.
x − ln x
11. f (x) = x sin x; 12. f (x) = x2 arccos x; 13. f (x) = x3 ex ; 14. f (x) = sin x cos x; 15. f (x) =
x2 + 3
x2 + 2x − 3
;
;
18.
f
(x)
=
x4
x3 − 2
Pochodne prostych złożeń
17. f (x) =
19. f (x) = x +
1
;
x+1
20. f (x) =
x + ln x
;
x − ln x
10
22. f (x) = sin 2x; 23. f (x) = e−x ; 24. f (x) = 1 + 2x2
; 25. f (x) = sin2 x; 26. f (x) = ln3 x; 27. f (x) = 4 ln (sin x) ;
√
√
2
28. f (x) = cos4 x; 29. f (x) = (1 − x)−15 ; 30. f (x) = e3x ; 31. f (x) = 1 − 3 cos x; 32. f (x) = ex ; 33. f (x) = 3x;
√
2
34. f (x) = esin x ; 35. f (x) = ln (2x) ; 36. f (x) = 3 ln (1 − 2x) ; 37. f (x) = 3e−3x ; 38. f (x) = 1 − x2 ;
39. f (x) = (1 + 3x)30 ;
40. f (x) = arctg 2x;
41. f (x) = arcsin x2 ;
Różne
x
3
42. f (x) = x3 ex ; 43. f (x) = x2 ln 1 + x2 ; 44. f (x) = xe1/x ; 45.f (x) = √
; 46. f (x) = x3 ln x2 + 1 + arctg x.
2
4−x
11x + 2x3
x x
5x
x2 ln x
; 48.f (x) = 21 arctg x +
;
49.
f
(x)
=
e
;
47. f (x) = −3 arctg x − 2
;
50.
f
(x)
=
x +1
x2 + 1
x +
1
x2 + 1
51. f (x) = 12 x2 − 2 ln x2 + 4 + 2 arctg x2 ; 52. f (x) = ln3 x2 + x + 1 ; 53. f (x) = sin2 x + 1;
√
2
x +1
√
xe
54. f (x) = sin x4 ; 55. f (x) = arccos 1 − x2 ; 56. f (x) = arctg 1 + x2 ; 57.∗ f (x) = √ 2
;
x
+
1
x
√
58.∗ f (x) = x sin2 3x + x ln x; 59.∗ f (x) = x2x ; 60.∗ f (x) = 1 + x1 ; 61.∗ f (x) = x 1 + sin x;
3
√
POCHODNE - ODPOWIEDZI
√
√
1. f ′ (x) = 4x − x22 − 9x2 ; 2. f ′ (x) = 12 − x12
2 + 2 ; 3. f ′ (x) = 8x + √12x ; 4. f ′ (x) = 6x + x22 + 12 ; 5. f ′ (x) = 32 x −
6. f ′ (x) = 2 cos x + 3 sin x; 7. f ′ (x) = ln 2 + x2 ; 8. f ′ (x) = 0; 9. f ′ (x) = 0; 10.f ′ (x) = cos12 x + sin12 x = sin2 x 1cos2 x .
2
x2
′
2 x
′
2
11. f ′ (x) = sin x + x cos x; 12. f ′ (x) = 2x arccos x − √1−x
2 ; 13. f (x) = x e (x + 3) ; 14. f (x) = cos x − sin x;
15. f ′ (x) =
20. f ′ (x) =
3√
;
2x2 x
2
3
1
6
−2x2 −12
−x4 −4
′
′
; 18. f ′ (x) = 9x −4x−4x
; 19. f ′ (x) = x(x+2)
;
x2 (1 − ln x) ; 16. f (x) = (x−3)2 ; 17. f (x) =
x5
(x3 −2)2
(x+1)2
2
9
2(1−ln x)
2x(1−2 ln x)
′
′
′
−x
′
; 21 f (x) = (ln x−x2 )2 . 22. f (x) = 2 cos 2x; 23. f (x) = −e ; 24. f (x) = 40x 2x + 1 ;
(x−ln x)2
−16
x
′
3
′
25. f ′ (x) = 2 cos x sin x; 26. f ′ (x) = x3 ln2 x; 27. f ′ (x) = 4 cos
;
sin x ; 28. f (x) = 4 cos x sin x; 29. f (x) = 15 (1 − x)
2
3
sin
x
3
′
3x
′
′
x
′
′
sin
x
′
30. f (x) = 3e ; 31. f (x) = √1−3 cos x ; 32. f (x) = 2xe ; 33. f (x) = 2√3x ; 34. f (x) = (cos x) e
; 35. f (x) = x1 ;
29
−x
1
′
f ′ (x) = −18xe−3x ; 38. f ′ (x) = √1−x
; 40. f ′ (x) = 4x22+1 ; 41. f ′ (x) = √4−x
2 ; 39. f (x) = 90 (3x + 1)
2.
3
3
1
x
x−1 x
4
′
x
′
2
′
′
√
42. f (x) = 3x x + 1 e ; 43. f (x) = 2x ln x + 1 + 2 x2 +1 ; f (x) = x e ; 45.f (x) = 4−x2 (4−x2 ) ;
2 2
2
4
2
2(x2 −4)
+1
x +4
+x+1 x
2x ln x+x3 +x
2
′
′
′
46. f ′ (x) = 2x
+
3x
ln
x
+
1
.
47.
f
(x)
=
;
48.
f
(x)
=
; 49. f ′ (x) = x(x+1)
;
2
2 e ; 50. f (x) =
2
2
x +1
1+x2
(x +1)
(x+1)2
2
2
x3 +4
2x+1
sin
x
cos
x
′
′
2
′
′
3
4
4
51. f (x) = x2 +4 . 52 f (x) = 3 x+x2 +1 ln x + x + 1 ; 53. f (x) = √ 2
; 54. f (x) = 12x cos x sin x ;
sin x+1
√
√
2
x2 +1
2
x
∗ ′
∗ ′
√1+x x +1 e
√ sin 3x cos 3x ;
;
58.
f
(x)
=
x ln x + sin2 3x+x ln x+1+6
55. f ′ (x) = |x|√x1−x2 ; 56. f ′ (x) = √x2 +1(x
2 +2) ; 57. f (x) =
2
2
1+x (x +1)
2 x ln x+sin2 3x
x
√
1
x cos x
59.∗ f ′ (x) = 2x2x (ln x + 1) ; 60.∗ f ′ (x) = x1 + 1
ln 1 + x1 − x+1
; 61.∗ f ′ (x) = x12 x 1 + sin x x1 1+sin
−
ln
(1
+
sin
x)
.
x
36. f ′ (x) =
−6
1−2x ; 37.
3
2
2
2