1. Tema 3. Derivadas.
“Soluciones cálculo de derivadas”
4 1
1. f ' ( x) = 9 x 2 + x −1+
3 3 x2
3
3 18
2. f ' ( x) = x 3 + 3 x + −
x2 x4
3 5 5
3. f ' ( x) = x− 3 +
2 2x x x 2 3 x 2
9 3
4. f ' ( x) = x− 4
10 10 x
5. ( ⎛
f ' ( x) = 2 x − x senx + ⎜ x 2 +
⎜ ) 1 ⎞
⎟ cos x
⎟
⎝ 2 x⎠
f ' ( x) = x 2 (1 + 3 ln x ) +
tan x 1
6. −
2x x x cos 2 x
3 x cot 2 x + 2 cot x + 3 x
7. f ' ( x) = − − ex
3 2
3x x
8. f ' ( x) = 2e x cos x
⎛ 1 ⎞
9. f ' ( x) = 4 x ⎜ ln 4 ⋅ arcsenx +
⎜
⎟
⎟
⎝ 1− x2 ⎠
1 1
10. f ' ( x) = arctan x + x
2 x 1+ x2
− 20 x 2 + 16 x − 5
11. f ' ( x) =
(4 x 2
−1 )2
2e x ( x − 1)
12. f ' ( x) =
(x − e )x 2
Fundamentos matemáticos en Arquitectura I Jesús Hernández Benito

2. x2
arcsenx − ( x − arctan x )
1
1+ x 2
1− x2
13. f ' ( x) =
(arcsenx )2
1
(1 − arctan x ) + x
2 x 1+ x2
14. f ' ( x) =
(1 − arctan x) 2
− 2 x + 1 − 3 ln x
15. f ' ( x) =
x4
−2
16. f ' ( x) =
1 − sen 2 x
17. f ' ( x) =
(2 + tan 2
)
x + cot 2 x x senx − (tan x − cot x )(senx + x cos x )
x 2 sen 2 x
2 x − 2 + ln x
18. f ' ( x) =
x2
(2 x arctan x + 1) ln x − 1 + x
2
arctan x
19. f ' ( x) = x
ln 2 x
20. f ' ( x) = e x [(1 + x ) senx + x cos x ]
21. f ' ( x) =
(3x senx + x
2 3
)
cos x ln x − x 2 senx
ln 2 x
1+ x
22. f ' ( x) = e x 2
2 x
23. (
f ' ( x) = 42 4 x 3 + 6 x − 2 ) (2 x
6 2
)
+1
2 x 3 − 3x
24. f ' ( x) =
x 4 − 3x 2 + 6
− 2x
25. f ' ( x) =
(
3 3 x2 − 5 )
4
Fundamentos matemáticos en Arquitectura I Jesús Hernández Benito

3. f ' ( x) = 5(senx − cos x ) (cos x + senx )
4
26.
2⎛ 3x ⎞
27. f ' ( x) = (arctan x ) ⎜ arctan x + ⎟
⎝ 1+ x2 ⎠
28. (
f ' ( x) = 1 − x 2 ) (arcsenx ) (3
4 2
1 − x 2 − 10 xarcsenx )
29. f ' ( x) = cot x
log10 e
30. f ' ( x) = cos x
2 x sen x
− x sen x − 2 cos x
f ' ( x) =
(x − cos x )
31. 2
−1
32. f ' ( x) =
1− x2
1 + sen10 x
33. f ' ( x) = −30
(1 − sen10 x) 2
−1
34. f ' ( x) =
x x2 −1
−1
35. f ' ( x) =
2 1− x2
− ex
36. f ' ( x) =
2e x − e 2 x
2
37. f ' ( x) =
x 2x 2 − 1
ex
38. f ' ( x) =
e2x − 1
1
arcsen
− ln 8 ⋅ 8 x
39. f ' ( x) =
x x2 −1
Fundamentos matemáticos en Arquitectura I Jesús Hernández Benito

4. sen 2 x
40. f ' ( x) =
1 − cos 4 x
−x
41. f ' ( x) =
2x − x 2
42. f ' ( x) = 2 a 2 − x 2
1
43. f ' ( x) =
ln 3 2ax + x 2
x +1
44. f ' ( x) =
x − a2
2
1 1
45. f ' ( x) = +
1 − x arcsenx
2
x 1 − ln 2 x
x a + a + x2
46. f ' ( x) =
a + x2
+
(
x a + x2 1+ a + x2 )
arcsenx
47. f ' ( x) =
(1 − x ) 2 3
1
48. f ' ( x) =
1+ x2
1 1 −2
49. f ' ( x) =
⎛ 1 − x ⎞ ln 1 − x 1 − x 2
ln⎜ ln ⎟
⎝ 1+ x ⎠ 1+ x
50. f ' ( x) = 8sen (sen 2 (sen 2 x )) ⋅ cos(sen 2 (sen 2 x )) ⋅ sen (sen 2 x ) ⋅ cos(sen 2 x ) ⋅ senx ⋅ cos x
⎛ 1 ⎞ 1
51. f ' ( x) = −sen⎜
⎜ arccos(senx ) ⎟ arccos 2 (senx )
⎟
⎝ ⎠
5 1 1
52. f ' ( x) = 2
9 ⎛5 x 4⎞ 2 x
1 + ⎜ tan + ⎟ cos 2
⎝3 2 3⎠
Fundamentos matemáticos en Arquitectura I Jesús Hernández Benito

5. ⎛ 11 ⎛ 1 ⎞⎞
53. f ' ( x) = ⎜1 + ⎜1 + ⎟⎟
⎜ 2 x+ x ⎜ ⎟
2 x+ x+ x ⎝ ⎝ 2 x ⎠⎟⎠
2 tan x(1 + tan 2 x )
54. f ' ( x) =
1 + tan 4 x
55. f ' ( x) = 3x 3 x (1 + ln x)
56. [
f ' ( x) = x x x x ln x(1 + ln x) + x x −1
x
]
1 x −1 ⎛ 1 ⎞
57. f ' ( x) = x ⎜1 + ln x ⎟
2 ⎝ 2 ⎠
f ' ( x) = x x +1
(1 + 2 ln x )
2
58.
1
⎛ 1 ⎞ senx ⎛ cos x 1 1 ⎞
59. f ' ( x ) = −⎜ ⎟ ⎜ ln + ⎟
⎝ x ⎠ ⎝ sen x x xsenx ⎠
2
Fundamentos matemáticos en Arquitectura I Jesús Hernández Benito