Serie 2: Derivadas

1) f(x) c) f(x)
2xt 3x 3x3 1x2
X3
7
-

=
-

+ - -
=

14x
y 18x + 9x 8x
-

=
-

y (2)(1) (x 3)(0)
-
-

12/2

y =

2 =

/
3) f(x) = - x+ x -
1 4) f(x) =
-

Ex Ex - +
2

y = (2)( 3x) -

+ 1) -

( x3 + x 1)(0)
- -

y =
-

33x 22x +

(2)2

y =
=
6x
4
+ 2
y (x= +
+ +

5) f(x) 6) f(x)
= t = 4x
-

=
+ -

Im (
L -

y=

=
y ((x3)(0)) (( C)(3xi)
j (x)(0) (3)(2x)
-

=
- -

=
3xz -
(3 (x + h)2)
.

(X3)2 (x )2
2

lim (x2)(x + h)2

h -
0 h

3xi -
(3x + 6xh + 3h2)
y = 6x
Xt
=

&
lim (x2)(x *
+2xh + hz)
h >0 h

y
-

3 - x2 6xh 3h2
= -

y +
3 -

1
S
-
-

lim xx + 2xh + x
h)
h-
-
lim-6xh-3h
:

h -
0 xh + 2xh2 + x243

lim hi -
bx -

3h)
h yo
h(x + 2xzh + ht)
-

=
x

lim -
6x -

3h =
-
6x -
310) = - 6x =-
↳
h -

>0 x + 2xh + x h) x + 2xi(0) + xi() XT x3 II

7) f(x) = X) -
1 U

(x 1)2
+ v
8) f(x) =
5x -

3x
3
U

X5 j

y xy =

x + 1

y' (x5)(20x 9x) (5x 3x9(5x)

-

=
-

(x5)2

y (xi + (x + 1)((x) (x
x)(2x + 2)
-
-

(x + 2x + 1)2
y' = 20xi -
9x7
X16
-

(25x* 15x) -

y' 2x + 4x + 2x (2x + 2x 2x 2)
-
-
-

(x2 1)2
y'
+ 2x + = 20x 9x7 -
-

25x + 15x7
16

y 2x3
-+ 4xi 2x7x X
= + -

2x + 2x + 2

(x
5x
+ 2x + 1) + 6x
y
-

2x + 4x +2
y
=

(x2 + 2x + 1)2
y 6)
=
= X ( -

5x +
x3
y' = 2(x2 + 2x + 1)
(x2 + 2x + 1)
= 5x + 6
y
-

X3
Il
y=
/
x3
y
=

1 /2
1
= & (x)
Y 2
9) f(x) 1 U
10) f(x) x3 3 x
y
-

3x
= =
=

X3v

(x (0))
y ()))]
- 3
x

I y
= =

((x) (x)
Y2
2(x)
Y

Y y
=

5
3x
y =

3 y =

2-
y
=

3x
2

y 3 y =
=

3 2
y
=- X -

233x
 11) f(x) =
-

3X -

23 ↑
5

uv
u v
=:

: x5
3X 23 X5
Y
-

y Y
=
-

= 3 (0) 2 5
- X2
y
X
- .
t
-

y
-
.

*
2x 3
)
y
=
-

103
3
X2
g
=

55(x
y -
=

2
3

513
xx
X

y
=

= 3 10 3 *
Y
- -

2x 3
I

(2) f(x) x
-x 15x
-

=
-
-

>
-
/
3
y
=
~

c
x
I .
15x
y y y
=
- = =

-* 3
(15)
y
=
((15x)
y
= [(
z))3x)) Y
-

3 X y =
3
x
y=
=

2
x
6
y
=
-

5x
15
y
=
-

x -
-

215X
Il
(3) +(x) =
-

-x3 -
2x5 -

5x

3
y -
=

c
*
(

= 10x
q
10x
y
-
x -
-

Il
 uv

14) f(x)
x y
a 3
X
y
=
= X =
X

v
= x. = 1 X 1
1 Y
. .

Y
.

x)(
1
3x)2x 2

y=
t

E
y 3x
3
3 X =

3
X
Y 2 X

= XY3
Y 3x"3
t

2 x/2

y =

x +

2 x6
y 23X +
=
3

j : +
=

x eX
y
=

-
y' E : x .
1

y =

-x)23x +3
-

xx)(2 y
=
1

z X

S
R
/X
2

2x + 3x
y
= ~

y
= 2 1(x)" (4)
Z ((x)*
y
= (1)

15) f(x) 2x 3 x =
= j 3.x
+
y 22
=
=

x
-X

y= [( x)1 ) 1
y= 2
-

+
3.x)
-

[(rx + +

2x 3 -
x

xx)z 2

-
y
=

Y2

= (x)

I
y
=

2x
16 f(x) = x
*
-

x3 + 3)9 17) f(x) =

(x2 2) -

4(x5 x + 3)5x 3x) 2(x 2 2)(2x)

y
=
- -

y =

20x4 12x2x5 x3 33 y
4x(x 2)
y
-

+ =
-
-

4x
y 8x/
-

18) f(x) = (x -
1)(xy1)2 y
=
(x + 1)

(x + 1)
y (X 1)(2x + 2)
y 2(x 1)(1)
=
-

+ 1 = +

2x2
y'
'

2x (x 1) (x +2
2x 2
y
= + -
-
+ +

2x 2 + x2 2x + 1
y =
-
+

y = 2x3 + 2x -

1/

f(x (x * *
+ 3)
(9) f(x) 3)9 20)
x
-

= (x5 x3 -

+ =

4(x5 x3 + 3)(5x 3x) (x5 x3 + 3) (5x

y
=

y + 3x)
-
- -
-

y
20xt 12x2)(x5 x3 +
333/ y 5x-3
- -

=
=

/
5x3 x3 + 3
Y
-

*
35
y y(x5 x3 (5x 3x2)
=
- + -

5
21) f(x) = x5 -

x3 +3
22) f(x) =
1
y = 5x1 -

3x
* - 55(x5 x3 + 3)9
35
y y(x5 (5x 3x2)
-

=
-
x3 + -

X x3 + 3-

y 5x1 3x
y
5x1 - x +30 1
3
- -

=
=

55(x5 x3
379/
-
+

2
(5x5 -
x3 + 3)

5x1 3x -

=
y 3
-
xi x + 3

5x5 x3 + 3)2-

=
Y z5X-3 3
I
23) f(x) 3 X5 X3a
+3 X5
x3
y
+
=
- -

V
X3

4 S

y z(x5 x3 +3 3x x3 6
y (xz)(5x 3x) (x* x3 +
3)(2x)]
- - -
- -
=
-

x X3
(x2))

y
= 3x5 x 6
y 5x 3x 2x + 2x 6x
-

-
= - - -

3x33 XI
XS x + 32 -

X
Il y' = 3x -

x 6x -

= X(3x - xi -
6)
X X(x3)

24) f(x) 5 xi
y=
= + X

X + 1
-

4
xi
y =(x =
+ y

+ 1 X Xy = [(x + 1)(2x +
1)] [(xi + x)(1)]
-

(x + 1)2

y' =

(5x
x + 4x + 1
y = 2x + x + 2x + 1 -
x + x

+ 5)(5xn
+
10x + x
x2 + 2x + 1
X + 1
Il
x + 4x +
y =

x2 + 2x + 1

* 25) f(x) = x + 2x + 1

Xz -

1
x2 + 2x + 1
-
12 y =

X2 1

y -(x)
-

= + 2x + 1
-
4x -
2

X2 -

1 (x2 -
1)2
=
y ((x2 1)(2x + 2)) [(x + 2x + 1)(2x)]
-
-

(x 1)2

y' =
2
-

x2 + 2x +1 1 3 *
2x #2x2 2x- 2 2x *
- & 2- 2x
Y - -

=
-
-

X2 -

1
(x2 =
1)2

4x 2 ! 2x= - 4x 2
y Y
-
-
- -

2 x2 + 2x + 1 (x" 1)2 - (x2 -

1)2
X2 1
Il
-

y =2

*
 26) f(x) = xi -
1
y
=

X
X2 2x -

+ 1

2) 4x ((x2 (x + 1)(2x)] [(x) 1)(2x 2)]

y' +
2x2
y
2
-

=
+
- -
- -
-

(x 2x + 1)2
-
(x2 -

2x + 1)2

'
2x 4x2 + 2x (2x3 2x 2x 2)
y
+
= 2x
- -

4x
-

2
-

y
-

+ -

2
exi (x * 2x + 1) (x2 2x + 1)2
-

/
y 2x= -

4x + 2x =x3 + 2x 2 2x -

(x2 -

2x + 1)2

y =
-

2x + 4x2
(x 2x + 1)2 -

* +
29) f(x)
+
27) f(x) 28) f(x) ex
* + 1
= e =
-

3 .

ex = 7 .

* *
3 ex
+ 2
ex +

y'
1
-1/
7 2X
y e
y
. ·
= =
-
= .

/ X2 + 1
14xe
y
=

//

ex
+ x 1

30) =(x) f(x)

-

3 31) ex 32) f(x) 7x

7
-

.
= = =

-
ex
+ x -
(2x + 1) Y (e zx - 1
(n(7)](2x)
y ((e y
3
y
=
- .

= =

* -
3ex
1
(2x)(7 1)
[(n(7)]/
+ x

y
-

6x
y
-

/
=
-
=

e
y=
ex

33) f(x) =
- 1

2x 2x
Y
I

2
y - *
[Ln(2)]
=
=2
Y
*
2 [Ln(2)] *
[)2x)(1)] [(1))22x
=

(24)
y
-

y
=

(2x(
* *** (2
' *
[Ln(21)
y +12
y xx
*

=
-

2 [\n(2))
2 2x
'=
*
2 Sun(2)]
2x y 2 2x
se
y - En
2x

y 2(2x) (Ln(2)]
*
=
-

(2 )(22x)
*

y =

@Cn = El
2x Il
34) f(x) = 2 ** 3 .
5x
* * *
= 2 [(n(2)](1) 3 5 [Ln (3 5)](1)
Y
-

. .

*
**
[2n (3 5))
y [In(2)]
= 2
-

3 5
/
.
.

35) f(x) (2 =
**
-

3 .
5x)3
5)))
*

5) (2 [Lu (3
** **

y = 3(2 -
3 .
(In (21] -

3 3
.
.

+ 1

36) f(x) = 34
+ 1
37)f(x) =
7x
(3x x +
[(n(7)]
y
7 1

Y =+
=
=

(3x
+
-* (z
**
[Ln (3)
2x +1

'= 1
7
**
[Ln(7)]
y 2x + 1 Il
+

[
gx 3
y
:
=

*
Il (x + 1) (1)
Y
1
y
=

2x + 1

e e ex2
3
38) f(x)
y
= e + = +

23x ex
y = 3 + 2x
.
-

* * * x2
(3)(3e
*
(] [(e +e 2xe
y y'
= + (xe -

((3)] = 3e +

13)2
* *

y = 9e + 6xe 3 3e
-
-

39) f(x)
=
*
((n(f)]((x))
*
[(x3)(7 [(7 ((3x4]
y
-

(x3)2

y
=
10) f(x) =

ex
*
)] ((e )(3xt)] *

y ((x3)(2xe
-

(x3)2
ex ex
y' 2x 3x
-

X
Il

7x
41) f(x)

g
=
X
* *
(((2x)(7 (((n(t))) ((7 ((3x4]
-

Y
2
* *

7x 2x ) [Ln(7)3
-

(7 -

(7 ((3x )
2
(x3)2
y *
=
X3 2x(z )((n(7)] (7 ((3x)
*

y'
-

X =

X
y (x (7xi)((n(7)]
*
=
-

(7 )(3x)
2 x6
7x I

(n(x 3)
y
<
= -

42) f(x) = (n(x + 3) 43) f(x) = 7x + (n(x 3) -

y X =

y =

x
+ 3 y = 7 +
y
/ /

44) f(x) = (n(x) 3x + 2) -

15)f(x)
=
y' bx 2(2x 3)
-

y +
=
(((n(x 1](0)] [(1)(x 1)]
y'
-

= -

= (In(X 1))2
y 25
-

/
y' -X-12
,
I
=

y =

(x -

1)[(n(x 1)72-

Il
 y X2 1
y =X
=

46) f(x) (nXi 1 -

X2 2x + 1 X2 2x +
- 1
[(x) (x + 1)(2x)] [(xi
y 1)(2x 2)]
-
= - -

y (x) 1)" (x" 2x + 1)

-

1
-

=
2x2 + 4x 2
y
-
-

= ·

X2 2x + 1
Exi
-

Xz - 1 (2 * +
)(X" -
2x + 1) y = 2x 4xi
- + 2x =
+ 2x + 2x -
1

(2x) *
(X2 -
2x + 1)

y t
X2 2x -
+ 1 =
-
xx2 + ex -
2

(x2 -

2x + 1)
!
y 2x2 + 4x 2
2x2 + 4x
-

2
-

Y
-
-

(x) 2x + 1)2
(2
-

)(x2 2x 1)
y= -C x
-

+
,
+

4)7) f(x) = Sen(x + 1) 18) f(x) = sen(2x + 2xi))

cos(2xi + 2x) (4xi + 4x)(6xi + (x)
y' =
cos(x + 1)(1)
y
=

//
cos(x + 1)
y =

/
(2x + 2x2)2
y=
=

2(2x + 2x)(6x + 4x)

Y (x + +xy)(6x + dx)
y
=

49) f(x) sen(x + 1) = + 5x 50) f(x) = (Sen(x + 1)

*
1) = (sen 1)] (cos(x + 1))
y *
cos(x
5/
+ (x +
y
= +

sen(x + 1) '= cos(x + 1)

y y
=

2
y = cs(x + 1) Sen(x + 1)
/

51) f(x) cos(3x +3) = 52) f(x) = cos(3x2 + 3x)

Sen(3xx + 3x)(6x
y =
-

Sen(3x + 3)(3)
y =
-
+ 3)

3 sen (3x + 3) 6x
y y = + 3sen(3xx +
3x)//
-

53) f(x) =

Sen (x + 1)

'=
y [sen(x 1)(0)] ((1)[cos(x 1)]]
+ -

ISen (x + 1))2

y' =

-
54) f(x) =
1 t
1
Sen(x
v
COS X + 1)
V

y= (cosx)(0)] [(1)( Sen(x)) - -

y = [(sen(x + 1)(0)] -

[(1)[cos(x + ))]

(Cos X)2 (Sen(x + 1)]2

cos(x + 1)
y
=

y Sex
-

[sen(x + 1)]

y =
Sen(x) -
cos(x + 1)

[cos(x1]2 [sen(x +
1)]2//
cos(x)
y
=

55) f(x) =
1
-
1 [sen(x)]2
Sen X Cos(X 1) '= 1)
-

Sen(x
y
L
-

[cos(x- 1)]2

y = cos(x)

[sen(x)]a
Sen(x

[cos(x-1)]2
-
1)

Il

56) f(x) = "(oc(3x + 3)

y- =
(cos(3x + 315-3( -

3sen(3x + 3)

y -3 sen 3x
=

II