Calclo II

Repaso : Derivadas de una

función de una variable
real -

X
y t ER
,
,
7
, ,
s
ejemplo
Y
-
f(x) = mx + b

V
-

m >
-

1 Q I
& &
or
X g
-

-
⑧

mc0

Q t(x) = ax + 6x + 2

I a) 0 ③ +xx =
eX

· ·
&

O O ⑧

..
8

a0

"
Defi :
Derivale de E2x ,
see fixs ran
funcion de variable

real
,
la derivada de Fax, que se
representa :

1) D[t2x]

4)t'(x)

(t()
3) =

↳ Operador
deriva

-
At =lin Ex S2

..
+
+
x)
*T

Bx ,
t(x
⑧
(x +
-
%
I
Y
2
/
Tf
-
=
m ,

XX +AX Xz -
XI

(xty
DX

- A
m =

*
 Derivadas
ejemplo
Derive hs
sig . funciones
3
4) f(x) = -

2x + 2x3 -
4x + z

x-xx
= -ex +

12 (x -s)
(x)-
= +

aX

-(x) 2(x)
-

(x
=

3
1 2(3)(x3 )
- -

= -
2( 3)(x -

+ -
4)X + 0

+
= 6x + 6x2 -
4Xo

4
6x2
-

=
6x + - 4

3x
-- 113
2)((xx = sx 3 + + = (xi + sx + 3)

+(x)
-
= (x + xx + 3) 3]
dx
23
=
-(x + sx + 35 .
(3x2 + s)

-
 "L2x)
got
*
ejemplo E(x) =
cos (e ) +

= Leos(e *
) + el
[ec 22x)]
+
= [cos" (e (]
* +

= ecot"
(2) Lot"]
+

-de
(xex2 ecot" (2(x)
d(dx((x)
= &
-
1 22x2
-

1 + 4X

Feecot)
2xex

1 + 4X
=
Ex12- x

=
2xe
*
cot
"
(2 *
x)
= x
C
=

Feixr
-

(1 +
1
 Derivadas

t(x)
El >
-

Anti-derivada

& Ex)(ax n
t(x)

S dx : antiderivada dx = DX -
>0

↳
↓ diferencial .

integral

JaxEcxi] : antiderivada de Ex

I
t(x)

-
x3
E
(N(2x) *
sen2(3x) 6 Sen (3x)(OS(3X)
 t(x t(x)

& f(x) X10

+
=
X
20 +/
X
E X2 ↳> f(x) = +X
20 + /
X2 3 21
↳ X T ↓
21
X

X3 *

i
X
xn +

nt R
X

jax[t'(xi] =
E (x)

Teorema
(dx[x] =
X** ; n +

(ax[x +
3 = 2x

ejemplo (dx[x] =

+ y =
1
32X32
=
=x3 +2

constante de S

integración
.
 (dx [
3 1416 + 1
ejemplo 310163 +
.

x =
X
3 1416
. +1

to
↑ 1416
x
=
4 . 1416
= 0 . 24X + C

Defi
: Se dice que una función FCX) es una antiderivada
·
primitiva de E2x) ,
si F'(x) = E(x) .

funcion : (2x)

anti-derivada : F(X)

· F(x) =
EX2 es anti-derivada de t(x1 = X

? F(x) X f(x)
por que
= =

·
F(x) = senx es primitiva de FIx =
coSX

por que
? #(x) = 205X = f(x)
· F(x) =
Ex76 +
1X ;
es
primitiva de

t(x) =

rx(iz + ) :

Ext- + x
F(x)
=

+ x( = " +

t()
= =
x113
+
13
y (2
-
-

0(x
+
=
2 p 0
+ y = x + x =

· F(x) =
+x = > ((x) =
X

Fy(x) =
(xi + 1 = > ((x) =
X

F(x) = X + 0 = X

Fe(X) =
EX2 + 5

Fz(x) =
+X2 -

114

In Unhiderivada General F(x1 =

Ext + C
↓

es
calquie
numero :

.
constante
Propiedades pasions

1(dx[a] = ax + c

2)(8x[t(x) g(x)) = =

(dx[t()] jax[g(x1] =

3)
(dx[at (x)) =
a)dx[t2]
ejemplo (dx [3x -
2x + X -

1]
=
3(dx[x3] -2
/dx[x2] + Jix[x]-)dx[1]
= 3
. - 1 .

x + 2

an a
Ext
-
m

= -

-x +
+ x - x +
2/ -

am
=

ejemplo Jax [ *
( +
*)]
Ja[ + ] (ax[
=
=

(dx[x(
13
x(x
-

4]
-

(ax [x" + x
(*
] =
Jax[x"0] (dx[x 0] +

1/4

-
+1
=4 =El
ejemplo
/dx[]
(dx [(xx)"
= - 2(3x((1) + (s)

X"3

Jax[ ] (dx[ ]
-
+
=
+

Sax[ax2 6x) x ]
13 3
-

=
-
+

a(dx[xs] 6)dx[x23] (dx[x ]

-

= 2/3 +1

X23
-

23
-

13 +/

=
-+
 Ejemplo
Jdx[ex-2cot
=

Jax[
=
3)dx[sax] 2(dx[] -

3)dx[senx] 2(dx[c0 + enx]

-

x
= -

3)dx[serx] -

2)dx[co + xcax]
posX
=
=
+
2/
+ 22xx

Ejemplo (dx[tanix + +
co 2x + 3]

(dx[saxx =
1 + asx -
1 + 3]

(dx[se2x + 2x2x + g]

Jax[seci] +
Jdx[csix] +
JUx[i
faux +
+
= -
co x + x
Seccion 2 Tecnica : Cambio de Variable .

Teorema seu funcion de variable real existe

gex) una

en intervalo I = Bu 81 sea texs funcion

y
un one
,
,

de variable real Continua en I entonces

y
.

Si Faxs es la Artiderivada de f(x

Jax[f(g(x))g'(x)) =
F(y(x) +C

>
- f(x s
eje-plo/dx[2 ·

2] =

L
↳
g(x) y'(x)
=
[((x 15 +

senti
↑[

ejemple (dx[ -
.
(x +
11]
-

g(x) y'(X)
cos(x x) +
1
=
-
+
 Ejemplo /dx[Exts'] =

Jex[ (2x + 3) "]

Cambio de Variable : X -
> U

& u = 2x + 3 = > u(x) = 2x + 3

②(2x3 (ax[(2x + 3) ]

- ①( =

([u]
③ du = 2 .
dx

= dx
=

Edu [a"]

ti
*
=
tu + 2

⑤ 22x 333
2
=
+ +