MA Mathematical analysis Derivatives 04.

2023

Here come the basic rules:

((af ± bg)(x))Õ = af Õ (x) ± bg Õ (x) (x– )Õ = –x–≠1 (– ”= 0)

(f (x)g(x))Õ = f Õ (x)g(x) + f (x)g Õ (x) (–x )Õ = ln – –x (0 < – ”= 1)
3 4 1
f (x) Õ f Õ (x)g(x) ≠ f (x)g Õ (x) (loga |x|)Õ = x ln (0 < – ”= 1)
= (sin x) = cos x
Õ
a
(cos x)Õ = ≠ sin x
g(x) g 2 (x)
1 1
((f ¶ g)(x)) = f (g(x)) · g Õ (x)
Õ Õ (arcsin x)Õ = Ô1≠x 2
(arccos x)Õ = ≠ Ô1≠x 2
1
–Õ = 0 (–—a constant) (arctan x)Õ = 1+x2
.

Exercise 1. In each case provide the derivative of the given function.

1 1
1. x ‘æ ex 28. x ‘æ 52. x ‘æ 2x sin x2 77. x ‘æ Ô 2
x ( x+1)
1
2. x ‘æ 2x3 ≠ 3x2 ≠ 7 29. x ‘æ x≠1 53. x ‘æ e 2
x
78. x ‘æ Ô
x
2
( x+1)
3. x ‘æ x100 ≠ x 30. x ‘æ 1
54. x ‘æ ln x1
x2 +1 x2
79. x ‘æ Ô 2
4. x ‘æ (x + 2)50 1 1 ( x+1)
31. x ‘æ x2 ≠1
55. x ‘æ ln x Ô
5. x ‘æ (x2 ≠3x+2)50 80. x ‘æ Ô
x
2
32. x ‘æ x
x+1 56. x ‘æ x ln 1
x
( x+1)
Ò
6. x ‘æ tan x 57. x ‘æ 1 81. x ‘æ
33. x ‘æ x+1 x+1
x ln x1 x≠2
7. x ‘æ cot x Ò
34. x ‘æ x≠1
x+1 58. x ‘æ sin x 82. x ‘æ x≠1
1≠sin x x+2
8. x ‘æ ln x sin x Ô
35. x ‘æ x+1
tan x 83. x ‘æ tan x
x≠1 59. x ‘æ tan x≠2
9. x ‘æ sin2 x
36. x ‘æ 2x≠1
sin x
84. x ‘æ arcsin2 x
x+2 60. x ‘æ
10. x ‘æ cos2 x 1≠cos x
85. x ‘æ 1
2x≠1
37. x ‘æ 1
arcsin x
11. x ‘æ tan2 x
2x+2 61. x ‘æ sin Ô
2x≠1
x
86. x ‘æ arccos x
38. x ‘æ 3x+1 x2 +1
12. x ‘æ sin x2 62. x ‘æ sin x≠2 87. x ‘æ arcsin(2 sin x)
x2 +1
39. x ‘æ 1
Ô
13. x ‘æ cos x2 x≠1 63. x ‘æ sin x
88. x ‘æ arcsin sin x
x2 ≠1
14. x ‘æ tan x2 40. x ‘æ x+2 64. x ‘æ 1 89. x ‘æ arcsin( fi2 tan x)
sin2 x
Ô
15. x ‘æ sin x cos x 41. x ‘æ x2 +1
65. x ‘æ x 90. x ‘æ arcsin x
x2 ≠1 1≠cos x

16. x ‘æ x sin x x2 +1
91. x ‘æ arcsin Ô1x
42. x ‘æ 2x2 ≠1 66. x ‘æ sin(x2 )
Ô
17. x ‘æ x2 sin x Ô 92. x ‘æ arctan x
43. x ‘æ x2 +1 67. x ‘æ cos x
2x2 ≠2
93. x ‘æ arctan Ô1x
18. x ‘æ x sin2 x Ô
68. x ‘æ x + 1
x2 +1
44. x ‘æ x2 ≠x≠1
Ô
19. x ‘æ sin x Ô 94. x ‘æ sin x
x
1+ x1 69. x ‘æ x2 + 1 2
45. x ‘æ 95. x ‘æ ln x≠1
x
20. x ‘æ x 1≠ x1
Ô1
sin x 70. x ‘æ
1+ x1
x≠1 96. x ‘æ ln cos x
x2
21. x ‘æ 46. x ‘æ
sin x x 71. x ‘æ Ôx
x≠1 97. x ‘æ ln sin x
sin x
22. x ‘æ x2
1+ 1
1+ x x
47. x ‘æ 72. x ‘æ Ô 1 98. x ‘æ sin ln x
x x2 +1
23. x ‘æ ex ln x
48. x ‘æ 1 99. x ‘æ cos ln x
2 1+ x1 73. x ‘æ Ô x
x2 +1
24. x ‘æ ln x 100. x ‘æ cos ln sin x
1 Ô
49. x ‘æ 74. x ‘æ x2 +1
25. x ‘æ x3 ln x 1+ 1 1
1+ x x 101. x ‘æ ln ln x
Ô 2
26. x ‘æ (ln x ≠ 1) ln x 50. x ‘æ sin 2x 75. x ‘æ ( x + 1) 102. x ‘æ sin(fi sin x)
Ô 2
27. x ‘æ x2 ex sin x 51. x ‘æ cos 3x 76. x ‘æ ( 3 x + 1) 103. x ‘æ cos(fi cos x)

1
 Ô
104. x ‘æ cos(fi sin x) 110. x ‘æ ln2 x 116. x ‘æ ln tan x2 122. x ‘æ logx x
Ô
105. x ‘æ ln(ex + 1) 111. x ‘æ ln sin x 117. x ‘æ ee 123. x ‘æ sinx x
x

2 +2
106. x ‘æ ex 112. x ‘æ esin ln x 118. x ‘æ xx 124. x ‘æ ln xx
Ô
107. x ‘æ e 113. x ‘æ ln cos ex 119. x ‘æ xx 125. x ‘æ (ln x)ln x
x≠1 x

Ô 1
108. x ‘æ eln x≠1 114. x ‘æ cos x2 120. x ‘æ x x 126. x ‘æ (sin x)sin x
Ô Ô Ô Ôx
109. x ‘æ ln x2 115. x ‘æ cos2 x 121. x ‘æ x 127. x ‘æ (sin x)ln x .

Exercise 2. Given the functions f , g and h determine the derivatives of the following
functions:

1 2 Ô
1. f 2 11. x ‘æ f 1
g
21. f 31. f f 41. f ¶ g ¶ f
! " Ô 1 2
‘ f x2
2. x æ 12. x ‘æ !11 " 22. g · f 32. ln ef + g 42. f ¶ f ¶ f ¶ f
f g Ò
3. f · g · h !g" 23. f
33. f ¶ g ¶ h 43. f ¶ f ¶ g ¶ g
13. x ‘æ f h
g
4. f ·g
Ô 44. f ¶ g ¶ f ¶ g
h
14. x ‘æ 1 24. f ¶ g 34. f g¶h
f( g
)
5. f Ô
h
g·h 25. f ¶ f 35. f g
h
45. f ¶ g ¶ g ¶ f
15. f ¶ ln
6. ln f 46. (f ¶ g)h
16. sin ¶f 26. ef 36. (ln f )g
7. ln ¶(f · g) 47. logf (g ¶ h)
17. f ¶ sin 27. f eg 37. f ¶ ln ¶g
1
8. f
18. f g 28. ef g 38. f ¶ f 48. logf ¶g h
1 2
1
9. x ‘æ f f
29. e g 39. f ¶ ln ¶f 49. ln f
x 19. logf g ln g
1 Ô
10. x ‘æ f ( x1 ) 20. x ‘æ f ( x) 30. ef ¶g 40. f ¶ f ¶ f 50. (logf g) ¶ h.