asdas

PÁGINA 294

222

SOLUCIONES
16. Quedan:

a) D ⎡ x 6 ⎤ = 6 x 5
⎣ ⎦
−35
⎡7⎤
b) D ⎢ 5 ⎥ = D ⎡7 x −5 ⎤ = − 7·5 x −6 = 6
⎣
⎦
x ⎦
x
⎣
1
2
c) D ⎡8 4 x ⎤ = D ⎡8 x 4 ⎤ = 4
⎣
⎦
⎢
⎥
⎣
⎦
x3

d) D ⎡3 x 2 − x + 4 ⎤ = 6 x − 1
⎣
⎦

(

)

(

4
e) D ⎡ x 2 + x ⎤ = ( 8 x + 4 ) x 2 + x
⎢
⎥
⎣
⎦

)

3

⎡ 5x ⎤
20
=
f) D ⎢
4 + 5 x ⎥ ( 4 + 5 x )2
⎣
⎦
⎡
1
g) D ⎢
⎢x5 − x2 + 3
⎣

(

)

⎤
⎥ = D ⎡ x5 − x2 + 3
5
⎢
⎥
⎣
⎦

(

)

−5

4
⎤ = −25 x + 10 x
⎥
⎦ x5 − x2 + 3 6

(

)

⎡ x 2 − 1⎤ x
h) D ⎢
⎥=
⎣ 4 ⎦ 2
⎡
⎤
3
⎥=
i) D ⎢
2
⎢
⎣ 4x + 5 ⎥
⎦

−12 x

( 4x

2

+5

)

3

17. Las derivadas quedan:
⎛ −3 ⎞
⎡ 3⎤
x
a) D ⎣ 4 x ⎦ = 4 ⋅ ln 4 ⋅ ⎜ 2 ⎟
⎝x ⎠
2
2
c) D ⎡e 2 x − e x − 2⎤ = e 2 x ⋅ 4 x − e x
⎣
⎦
⎡ e−2 x ⎤ −e −2 x
e) D ⎢
⎥=
2
⎣ 4 ⎦
3

b) D ⎡3 ⋅ 2 x ⎤ = 3 ⋅ 2 x ⋅ ln 2
⎣
⎦
d) D ⎡2 x ⋅ 3 x ⎤ = D ⎡ 6 x ⎤ = 6 x ⋅ 2 x ⋅ ln 6
⎣
⎦
⎣ ⎦
2

(

2

2

)

2

(

)

3
2
f) D ⎡ e 2 x + 1 ⎤ = 6 ⋅ e 2 x e 2 x + 1
⎢
⎥
⎣
⎦

223

18. Las derivadas quedan:

(

)

2x
a) D ⎡ln x 2 + 7 ⎤ = 2
⎣
⎦ x +7

(

x

)

e
b) D ⎡ln e x + 2 ⎤ = x
⎣
⎦ e +2

(

c) D ⎡ln3 − 4 x 3
⎣

)

5

2
2
⎤ = D ⎡5 ⋅ ln 3 − 4 x 3 ⎤ = 5 ⋅ −12 x 3 = −60 x 3
⎦
⎣
⎦
3 − 4x 3 − 4x

(

)

2x
⎡1
⎤ 1 6x
= 2
d) D ⎡ln 3 3 x 2 + 1⎤ = D ⎢ ln 3 x 2 + 1 ⎥ = · 2
⎣
⎦
⎣3
⎦ 3 3x + 1 3x + 1

(

(

)

)

2x
e) D ⎡log2 x 2 + 1 ⎤ = 2
⎣
⎦ x + 1 ln 2

(

)

−

1

1

⎡ 1− x ⎤
− x
2 x 2 x
f) D ⎢ln
−
=
⎥ = D ⎡ln 1− x − ln 1+ x ⎤ =
⎣
⎦ 1− x 1 + xx − x 2
⎢ 1+ x ⎥
⎣
⎦

(

) (

)

1
1
g) D ⎡ln ( ln x ) ⎤ = x =
⎣
⎦ ln x x ⋅ ln x

(

)

1
4 − 2x 2
⎡
⎤ 1 1 −2 x
=
h) D ⎡ln x ⋅ 4 − x 2 ⎤ = D ⎢ln x + ln 4 − x 2 ⎥ = + ⋅
2
⎢
⎥
⎣
⎦
2
x 4 − x2
⎣
⎦ x 2 4−x

(

)

(

)

⎡ 1+ x ⎤
−1
1
2
i) D ⎢ln
⎥ = D ⎡ln (1+ x ) − ln (1− x ) ⎤ = 1+ x − 1− x = 1− x 2
⎣
⎦
⎣ 1− x ⎦
19. Las derivadas quedan:
a) D [ sen4 x ] = 4 ⋅ cos 4 x
b) D [ 4 sen x ] = 4 ⋅ cos x
⎡
x
⎛ x ⎞⎤ 1
c) D ⎢sen ⎜ ⎟ ⎥ = cos
4
⎝ 4 ⎠⎦ 4
⎣

224

⎡
4
⎛ 4 ⎞⎤
⎛4⎞
d) D ⎢ sen ⎜ ⎟ ⎥ = − 2 ⋅ cos ⎜ ⎟
x ⎠⎦
x
⎝
⎝x⎠
⎣
e) D ⎡ sen x 4 ⎤ = 4 x 3 ⋅ cos x 4
⎣
⎦
4
f) D ⎡ sen4 x ⎤ = D ⎡( sen x ) ⎤ = 4 ⋅ sen3 x ⋅ cos x
⎣
⎦
⎣
⎦

g) D ⎡arc sen x − 1⎤ =
⎣
⎦

h) D ⎡sen x ⎤ =
⎣
⎦
−4

i) D ⎡ 4 sen x ⎤ =
⎣
⎦

1
2 ⋅3x − 2 − x 2

( )

−4 ⋅ cos x −4
x

5

cos x
4 4 sen3 x

j) D ⎡cos ( x + 1) ⎤ = − sen (x + 1)
⎣
⎦

(

)

(

)

(

)

k) D ⎡cos3 x 3 + 1 ⎤ = − 9 x 2 ⋅ cos2 x 3 + 1 ⋅ sen x 3 + 1
⎣
⎦
l) D ⎡arc tg (2 x + 1)2 ⎤ =
⎣
⎦

(

)

4x + 2
8 x + 16 x 3 + 12 x 2 + 4 x + 1
4

2x
m) D ⎡ tg x 2 + 2 ⎤ =
⎣
⎦ cos2 x 2 + 2

(

)

n) D tg x =

(

)

1
2 x ⋅cos2 x

3 tg2 ( x + 1)
ñ) D ⎡ tg3 ( x + 1)⎤ =
⎣
⎦ cos2 ( x + 1)
o) D [arc cos (ln x )] =

−1
x ⋅ 1− ln2 x

3 x ⋅ ln 3
p) D ⎡ tg (3 x )⎤ =
⎣
⎦ cos2 3 x
1
q) D ⎡ tg x ⎤ =
⎣
⎦ 2 cos2 x ⋅ tg x

225

20. Las derivadas quedan:
a) D [f (x)] =

2x 4 − 1
x 2 1+ 2 x 2

⇒ D [f (1)] =

1
3

c) D [ h (x)] = 12 sen 3 x ⋅ cos 3 x ⇒ D [ h (π)] = 0

b) D [ g (x)] =
d) D [ j (x)] =1
4 + x2
2 x ⋅ ln 2

(2

x

)

+1

2

⇒ D [ g (0)] =

1
2

⇒ D [ j ( − 1)] =

2 ln 2
9

21. El estudio en cada caso queda:
•

f ( x ) = 2x 4 + 3 x 3 + x 2 − ax + 5 ⇒ D [f ( x )] = 8 x 3 + 9 x 2 + 2x − a ⇒ D [f (1)] = 19 − a = − 3 ⇒ a = 22

•

g(x ) =

x2 − x − a
x 2 + 2x + a − 1
2+a
⇒ D [g ( x )] =
⇒ D [g (1)] =
=0 ⇒ a=− 2
2
x +1
4
( x + 1)

226PÁGINA 295

227

SOLUCIONES
22. Las derivadas quedan:

a) D ⎡(1− x ) 1+ x ⎤ = − 1 1+ x +
⎣
⎦

1− x
2 1+ x

=

−1− 3 x
2 1+ x

b) D ⎡( x 2 − 1) ⋅ 52 x ⎤ = 2 x ⋅ 52 x + 52 x ⋅ ln 5 ⋅ 2 ⋅ ( x 2 − 1)
⎣
⎦
c) D ⎡2 x ⋅ ln 2⎤ = 2 x ⋅ (ln 2)2
⎣
⎦
⎡ ⎛ 1+ sen x ⎞ ⎤
−cos x
cos x
2
−
=
d) D ⎢ln ⎜
⎟⎥ =
⎣ ⎝ 1− sen x ⎠ ⎦ 1+ sen x 1− sen x cos x
e) D ⎡ x 2 ln x + x ln x 2 ⎤ =...