demostraciones de derivadas por medio de limites
DESCRIPTION
Este documento, es una guía para el estudiante que dese saber como obtener las formulas de las derivadas usuales. Mediante un lenguaje formal se demuestran una a una las derivadas, empleando la definición formal.TRANSCRIPT
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
Demuestre empleando la definición formal de derivada, cada una de las siguientes expresiones:
D V N Ing. ELECTICA POTENCIA Página 1
13 .)[ sen ( x ) ]'=cos ( x ) .14 . )[cos ( x ) ] '=−sen (x ).15 .)[ tg ( x ) ]'=sec2( x ).16 .) [ctg( x ) ] '=−csc2( x ) .17 .) [sec ( x ) ]'=sec ( x )⋅tg ( x ) .18 .)[ csc ( x ) ]'=−csc ( x )⋅ctg( x ).
19 .)[ arcsen( x ) ] '=1
√1−x2.
(−1<x<1) .
20 .)[ arccos( x ) ] '=−1
√1−x2.
(−1<x<1) .
21 .)[ arctg( x )] '=11+x2
.
22 .)[ arcctg (x )] '=−11+x2
.
23 .)[ arc sec ( x ) ] '=1|x|⋅√ x2−1
,(1<|x|) .
24 . )[ arc csc( x )] '=−1|x|⋅√x2−1
,(1<|x|).
1. )(x )'=1 .2. )(k ) '=0 .3. )( xn) '=nxn−1.4 . )(ax) '=ax ln( a) .5. )(ex )'=e x .
6 .)[ ln( x ) ] '=1x,( x>0 ).
7 .)[ loga ( x ) ]'=1x ln(a )
≡loga (e )x
.
( x>0 )∧(a>0 )8 .)[ f ( x )±g ( x ) ]'=f '( x )±g ' ( x ).9 .)[ f ( x )⋅g (x )] '=f ' ( x )⋅g( x )+ f ( x )⋅g ' ( x ).
10. )[ f ( x )g ( x ) ]'
=f '( x )⋅g ( x )−f ( x )⋅g '( x )
[g (x )]2g ( x )≠0 .
11. )[ k⋅f ( x ) ]'=k⋅f ' ( x )
12. )[kf ( x ) ]'
=−k
[ f (x )]2, f ( x )≠0 .
2 .) f ( x )=k⇒ k≡x0k∴ f ( x )= x0 kf ( x+h )=( x+h )0 k
( k ) '=limh→0
( x+h )0k−x0kh
≡k limh→0
( x+h )0−x0
h
( k ) '=k limh→0
x+hx+h
−xx
h≡k lim
h→0
x (x+h )−x ( x+h )x ( x+h)h
( k ) '=k limh→0
x ( x+h ) (1−1 )x (x+h )h
( k ) '=k limh→0
(1−1 )+(eh−eh )h
( k ) '=k limh→0
(eh−1 )−(eh−1 )h
( k ) '=k [ limh→0 (eh−1 )h
−limh→0
(eh−1 )h ]
( k ) '=k (1−1 ) ∴( k ) '=0R // .
1 .) f ( x )=x
[ f ( x ) ] '=limh→0
f ( x+h )−f ( x )h
.
( x )'=limh→ 0
( x+h )−xh
( x )'=limh→ 0
x+h−xh
( x )'=limh→ 0
hh=1 .R //
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 2
3 .) f (x )=xn
( xn ) '= limx→0
( x+h )n−xn
h
( xn ) '= limh→0
(n0 )xn+(n1 ) xn−1h+(n2 ) xn−2 h2+…+(nn−2) x2hn−2+(nn−1)xhn−1+(nn )hn−xn
h
( xn ) '= limh→0
xn+nxn−1h+n(n−1)2
xn−2 h2+…+n (n−1 )2
x2hn−2+nxhn−1+hn−xn
h
( xn ) '= limh→0
nxn−1h+n(n−1)2
xn−2h2+…+n(n−1)2
x2hn−2+nxhn−1+hn
h
( xn ) '= limh→0
h[nxn−1+n(n−1)2xn−2h+…+
n(n−1)2
x2hn−3+nxhn− 2+hn−1]h
( xn ) '= limh→0 [nxn−1+n(n−1)2
xn−2 h+…+n(n−1)2
x2hn−3+nxhn−2+hn−1]( xn ) '=nxn−1+
n(n−1)2
xn−2( 0)+…+n (n−1 )2
x2 (0)+nx (0 )+(0 )
( xn ) '=nxn−1R // .
( ax) '=ax ln (a ) limt→0
(1t )(1t )
⋅tln ( t+1 )
( ax) '=ax ln (a ) limt→0
11tln ( t+1)
( ax) '=ax ln (a ) limt→0
1
ln ( t+1)1t
( ax) '=ax ln (a )1
limt→0ln ( t+1)
1t
limt→0ln ( t+1 )
1t ≡ln [ lim
t→0( t+1)
1t ]=ln (e )
( ax) '=ax ln (a )R // .
4 . ) f ( x )=ax
(ax )'=limh→0
ax+h−ax
h.
(ax )'=limh→0
ax (ah−1 )h
(ax )'=ax limh→0
ah−1h
t=ah−1⇒h=ln( t+1)ln( a)
(ax )'=ax limt→ 0
tln( t+1 )ln(a )
(ax )'=ax limt→ 0
ln(a )tln( t+1 )
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 3
5 .) f (x )=ex
( ex )'=limh→0
ex+h−ex
h.
( ex )'=limh→0
ex (eh−1 )h
( ex )'=ex limh→0
eh−1h
t=eh−1⇒h=ln( t+1)ln(e )
( ex )'=ex limt→0
tln( t+1 )
( ex )'=ex limt→0
11tln ( t+1 )
( ex )'=ex1
limt→0ln( t+1 )
1t
limt→0ln( t+1 )
1t ≡ln [ lim
t→0( t+1)
1t ]=ln (e )
( ex )'=ex R // .
6 .) f ( x )=ln( x )
[ ln (x )] '=limh→0
ln (x+h )−ln( x )h
[ ln (x )] '=limh→0
ln (x+hx )h
[ ln (x )] '=limh→0
1hln(1+hx )
[ ln (x )] '=limh→0
hx⋅1h⋅[ xh ln(1+hx )]
[ ln (x )] '=limh→0
1x⋅[ ln(1+hx )
xh ]
[ ln (x )] '=1xlimh→0 [ ln(1+hx )
xh ]
limh→0 [ ln(1+hx )
xh ]≡ln [ limh→0(1+hx )
xh ]=ln(e )
[ ln (x )] '=1xR // .
7 .) f ( x )=loga( x )
[ log a( x ) ] '=limh→0
loga ( x+h)−loga ( x )h
[ log a( x ) ] '=limh→0
loga (x+hx )h
[ log a( x ) ] '=limh→0
1hlog a(1+hx )
[ log a( x ) ] '=limh→0
hx⋅1h⋅[ xh loga (1+hx )]
[ log a( x ) ] '=1x⋅limh→0 [ loga(1+hx )
xh ]
limh→0 [ loga (1+hx )
xh ]≡log a[ limh→0 (1+hx )
xh ]=loga (e )
loga(e )≡ln (e )ln ( a)
∴[ loga( x )] '=1x ln (a )
R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 4
8 .) Sean f ( x ) yg( x )diferenciables enun Intervalo I .Hallar : [ f ( x )+g( x ) ] ' y [ f ( x )−g ( x ) ]'
[ f ( x )+g( x ) ] '=limh→0
[ f ( x+h )+g (x+h )]−[ f ( x )+g( x )]h
[ f ( x )+g( x ) ] '=limh→0
[ f ( x+h )−f (x )]+ [g (x+h )−g (x )]h
[ f ( x )+g( x ) ] '=limh→0 [ f ( x+h )−f ( x )
h+g( x+h )−g( x )h ]
[ f ( x )+g( x ) ] '=limh→0
f ( x+h )−f (x )h
+limh→0
g( x+h)−g( x )h
[ f ( x )+g( x ) ] '=[ f ( x ) ] '+[ g( x ) ] ' R // .
[ f ( x )−g( x ) ] '=limh→0
[ f ( x+h )−g( x+h )]−[ f ( x )−g( x ) ]h
[ f ( x )−g( x ) ] '=limh→0
[ f ( x+h )−f ( x )]−[ g( x+h )−g( x ) ]h
[ f ( x )−g( x ) ] '=limh→0 [ f ( x+h)−f ( x )
h−g( x+h)−g( x )h ]
[ f ( x )−g( x ) ] '=limh→0
f (x+h )−f ( x )h
−limh→0
g ( x+h)−g ( x )h
[ f ( x )−g( x ) ] '=[ f ( x )] '−[ g( x ) ] ' R // .
9 .) Sean f ( x ) yg( x )diferenciables enun Intervalo I .Hallar : [ f ( x )⋅g( x ) ] '
[ f ( x )⋅g( x ) ] '=limh→0
[ f ( x+h )⋅g( x+h) ]−[ f ( x )⋅g ( x )]h
[ f ( x )⋅g( x ) ] '=limh→0
[ f ( x+h )⋅g( x+h) ]−[ f ( x )⋅g ( x )]+ [ f ( x )⋅g ( x+h)−f ( x )⋅g (x+h )]h
[ f ( x )⋅g( x ) ] '=limh→0
[ f ( x+h )⋅g( x+h)−f ( x )⋅g ( x+h) ]+[ f ( x )⋅g( x+h )−f ( x )⋅g( x )]h
[ f ( x )⋅g( x ) ] '=limh→0
g ( x+h)⋅[ f ( x+h)−f ( x )]+ f ( x )⋅[g( x+h )−g( x )]h
[ f ( x )⋅g( x ) ] '=limh→0 [ g( x+h)⋅f ( x+h )−f ( x )
h+ f ( x )⋅
g( x+h )−g( x )h ]
[ f ( x )⋅g( x ) ] '=limh→0
g ( x+h)⋅limh→ 0
f ( x+h )−f (x )h
+limh→0
f ( x )⋅limh→0
g ( x+h)−g ( x )h
[ f ( x )⋅g( x ) ] '=g ( x+0)⋅[ f ( x ) ] '+ f ( x )[ g (x )] '[ f ( x )⋅g( x ) ] '=g ( x )⋅[ f ( x ) ]'+f ( x )[ g( x ) ] ' R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 5
10 . )Sean f ( x ) yg (x )diferenciables enun Intervalo I .
Hallar :[ f (x )g( x ) ]'
[ f ( x )g ( x ) ]
'
=limh→0
f ( x+h )g ( x+h)
−f ( x )g ( x )
h
[ f ( x )g ( x ) ]
'
=limh→0
f ( x+h )⋅g( x )−f ( x )⋅g (x+h )h⋅[ g( x+h )⋅g( x )]
[ f ( x )g ( x ) ]
'
=limh→0
[ f ( x+h )⋅g( x )−f ( x )⋅g (x+h )]+[ f (x )⋅g ( x )−f ( x )⋅g( x )]h⋅[ g( x+h )⋅g( x )]
[ f ( x )g ( x ) ]
'
=limh→0
[ f ( x+h )⋅g( x )−f ( x )⋅g (x )]−[ f (x )⋅g( x+h)−f ( x )⋅g ( x )]h⋅[ g( x+h )⋅g( x )]
[ f ( x )g ( x ) ]
'
=limh→0
g ( x )⋅[ f ( x+h)−f ( x )]−f ( x )⋅[g (x+h )−g( x )]h⋅[ g( x+h )⋅g( x )]
[ f ( x )g ( x ) ]
'
=limh→0 [ g( x )g( x+h )⋅g ( x )
⋅f ( x+h)−f ( x )h
−f ( x )g( x+h )⋅g ( x )
⋅g( x+h )−g( x )h ]
[ f ( x )g ( x ) ]
'
=limh→0
g ( x )g ( x+h)⋅g (x )
⋅limh→0
f ( x+h)−f ( x )h
+ limh→ 0
f ( x )g (x+h )⋅g( x )
⋅limh→0
g ( x+h)−g ( x )h
[ f ( x )g ( x ) ]
'
=g ( x )g ( x+0)⋅g (x )
⋅[ f (x )] '−f ( x )g( x+0 )⋅g( x )
⋅[ g( x )] '
[ f ( x )g ( x ) ]
'
=g ( x )⋅[ f ( x ) ]'g ( x )⋅g( x )
−f ( x )⋅[ g( x )] 'g( x )⋅g ( x )
≡g( x )⋅[ f ( x ) ] '−f ( x )⋅[ g (x )] '
[ g( x )]2R // , g( x )≠0
11. )Sea g( x )diferenciable enun Intervalo I .
Hallar :[kg( x ) ]' , k∈ℜ .
[kg ( x ) ]'
=limh→0
kg ( x+h)
−kg ( x )
h≡[kg( x ) ]
'
=limh→0
k⋅g ( x )−k⋅g( x+h)h⋅[g (x+h )⋅g( x ) ]
[kg ( x ) ]'
=limh→0
−k⋅[g( x+h )−g( x )]h⋅[ g( x+h )⋅g( x )]
≡[kg( x ) ]'
=limh→0
−kg( x+h )⋅g( x )
⋅limh→0
g (x+h )−g (x )h
[kg ( x ) ]'
=−kg ( x+0)⋅g (x )
⋅[ g( x ) ] '≡[kg ( x ) ]'
=−k⋅[ g (x )] '[ g( x ) ]2
R // , g( x )≠0 .
12 .) Sea f ( x )diferenciable enun Intervalo I .Hallar :[ k⋅f ( x ) ] ',k∈ℜ .
[ k⋅f ( x ) ]'=limh→0
k⋅f (x+h )−k⋅f ( x )h
[ k⋅f ( x ) ]'=limh→0
k⋅[ f (x+h )−f ( x )]h
[ k⋅f ( x ) ]'=k⋅limh→0
f (x+h )−f ( x )h
[ k⋅f ( x ) ]'=k⋅[ f (x )] ' R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
Derivadas de funciones trigonométricas y sus inversas.
D V N Ing. ELECTICA POTENCIA Página 6
13 . )Sea f ( x )=sen ( x )Hallar [ f ( x )] ' .
[ sen (x )] '=limh→0
sen( x+h )−sen ( x )h
[ sen (x )] '=limh→0
sen( x )cos (h)+sen(h )cos( x )−sen ( x )h
[ sen (x )] '=limh→0
sen(h )cos ( x )h
+ limh→0
sen( x )cos (h )−sen ( x )h
[ sen (x )] '=cos ( x )⋅limh→0
sen(h )h
+limh→0
sen ( x )⋅[cos (h )−1 ]h
[ sen (x )] '=cos ( x )⋅(1 )+ limh→0
sen ( x )⋅[cos (h)−1 ]h
⋅[cos (h)+1 ][cos (h)+1 ]
[ sen (x )] '=cos ( x )+limh→0
[cos2(h )−1 ]h
⋅sen (x )cos(h )+1
[ sen (x )] '=cos ( x )−limh→0
sen2( h)h
⋅limh→ 0
sen (x )cos(h )+1
[ sen (x )] '=cos ( x )−limh→0
sen(h )h
⋅limh→ 0
sen(h )sen ( x )cos (h )+1
[ sen (x )] '=cos ( x )−(1 )⋅sen(0 )sen (x )cos (0 )+1
[ sen (x )] '=cos ( x )−(0)⋅sen (x )1+1
[ sen (x )] '=cos ( x )−02
[ sen (x )] '=cos ( x )R // .
14 .)Sea f ( x )=cos ( x )Hallar [ f ( x )] ' .
[ cos( x ) ] '=limh→0
cos( x+h )−cos ( x )h
[ cos( x ) ] '=limh→0
cos( x )cos (h )−sen (h )sen( x )−cos (x )h
[ cos( x ) ] '=limh→0
cos( x )cos (h )−cos ( x )h
−limh→0
sen(h )sen ( x )h
[ cos( x ) ] '=limh→0
cos( x )⋅[cos(h )−1 ]h
−sen ( x )⋅limh→0
sen(h )h
[ cos( x ) ] '=limh→0
cos( x )⋅[cos(h )−1 ]h
⋅[cos(h )+1 ][cos(h )+1 ]
−sen (x )⋅(1)
[ cos( x ) ] '=limh→0
[cos2(h )−1 ]h
⋅cos ( x )cos (h )+1
−sen( x )
[ cos( x ) ] '=−sen ( x )−limh→0
sen2 (h)h
⋅limh→0
cos( x )cos(h )+1
[ cos( x ) ] '=−sen ( x )−limh→0
sen(h )h
⋅limh→0
sen( h)cos (x )cos(h )+1
[ cos( x ) ] '=−sen ( x )−(1)⋅sen(0 )cos( x )cos (0 )+1
[ cos( x ) ] '=−sen ( x )−02
[ cos( x ) ] '=−sen ( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 7
15 . )Sea f ( x )=tg ( x )Hallar [ f ( x ) ] ' .
[ tg ( x ) ] '=limh→0
tg (x+h)−tg ( x )h
[ tg ( x ) ] '=limh→0
sen ( x+h)cos (x+h )
−sen ( x )cos (x )
h
[ tg ( x ) ] '=limh→0
sen ( x+h)cos ( x )−sen( x )cos ( x+h)h⋅[cos ( x+h)⋅cos( x ) ]
;α=x+h , β=x .
sen (α−β )=sen(α )cos ( β )−sen( β )cos (α )
[ tg ( x ) ] '=limh→0
sen ( x+h−x )h⋅[cos ( x+h)⋅cos( x ) ]
[ tg ( x ) ] '=limh→0
sen (h )h
⋅limh→0
1cos (x+h )⋅cos ( x )
[ tg ( x ) ] '=(1)⋅1cos( x+0 )⋅cos ( x )
[ tg ( x ) ] '=1cos (x )⋅cos (x )
≡1cos2( x )
=sec2( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 8
15 . )Sea f ( x )=tg ( x )Hallar [ f ( x ) ] ' .
[ tg ( x ) ] '=limh→0
tg (x+h)−tg ( x )h
[ tg ( x ) ] '=limh→0
sen ( x+h)cos (x+h )
−sen ( x )cos (x )
h
[ tg ( x ) ] '=limh→0
sen ( x+h)cos ( x )−sen( x )cos ( x+h)h⋅[cos ( x+h)⋅cos( x ) ]
;α=x+h , β=x .
sen (α−β )=sen(α )cos ( β )−sen( β )cos (α )
[ tg ( x ) ] '=limh→0
sen ( x+h−x )h⋅[cos ( x+h)⋅cos( x ) ]
[ tg ( x ) ] '=limh→0
sen (h )h
⋅limh→0
1cos (x+h )⋅cos ( x )
[ tg ( x ) ] '=(1)⋅1cos( x+0 )⋅cos ( x )
[ tg ( x ) ] '=1cos (x )⋅cos (x )
≡1cos2( x )
=sec2( x )R // .
16 . )Sea f (x )=ctg( x )Hallar [ f ( x ) ]' .
[ ctg(x )] '=limh→0
ctg( x+h )−ctg( x )h
[ ctg(x )] '=limh→0
cos ( x+h)sen( x+h )
−cos ( x )sen( x )
h
[ ctg(x )] '=limh→0
sen( x )cos ( x+h)−sen (x+h )cos( x )h⋅[sen ( x+h)⋅sen( x ) ]
;α=x , β=x+h.
sen (α−β )=sen(α )cos ( β )−sen( β )cos (α )
[ ctg(x )] '=limh→0
sen( x−h−x )h⋅[sen ( x+h)⋅sen( x ) ]
[ ctg(x )] '=limh→0
−sen (h )h
⋅limh→0
1sen ( x+h )⋅sen ( x )
[ ctg(x )] '=(−1 )⋅1sen ( x+0)⋅sen( x )
[ ctg(x )] '=−1sen( x )⋅sen (x )
≡−1sen2 ( x )
=−csc2 ( x )R // .
17 . )Sea f (x )=sec( x )Hallar [ f ( x ) ] ' .
[ sec( x ) ] '=limh→0
sec( x+h)−sec( x )h
[ sec( x ) ] '=limh→0
1cos(x+h )
−1cos( x )
h
[ sec( x ) ] '=limh→0
cos(x )−cos ( x+h)h⋅[cos( x+h)⋅cos( x ) ]
[ sec( x ) ] '=limh→0
cos(x )−cos ( x )cos(h )+sen ( x )sen (h)h⋅[cos( x+h)⋅cos( x ) ]
[ sec( x ) ] '=limh→0
cos(x )⋅[1−cos (h )]h⋅[cos( x+h)⋅cos( x ) ]
+ limh→ 0
sen (x )sen( h)h⋅[cos ( x+h)⋅cos ( x )]
[ sec( x ) ] '=limh→0
cos(x )⋅[1−cos (h )]h⋅[cos( x+h)⋅cos( x ) ]
⋅[1+cos(h )][1+cos(h )]
+ limh→0
sen(h )h⋅¿
⋅limh→0
sen( x )[cos( x+h)⋅cos( x ) ]
¿
[ sec( x ) ] '=limh→0
[1−cos2(h )]h
⋅limh→ 0
cos( x )[cos(h )+1 ]⋅[cos(x+h )⋅cos ( x )]
+(1 )⋅sen ( x )cos ( x+0)⋅cos( x )
[ sec( x ) ] '=limh→0
sen2 (h )h
⋅limh→0
cos( x )[cos(h)+1 ]⋅[cos( x+h )⋅cos(x )]
+sen ( x )cos( x )⋅cos( x )
[ sec( x ) ] '=limh→0
sen (h )h
⋅limh→0
sen (h )cos ( x )[cos (h)+1 ]⋅[cos( x+h)⋅cos( x ) ]
+1cos( x )
⋅sen ( x )cos( x )
[ sec( x ) ] '=(1 )⋅sen (0)cos( x )[cos(0 )+1 ]⋅[cos ( x+0)⋅cos ( x )]
+sec( x )⋅tg( x )
[ sec( x ) ] '=02cos2 (x )
+sec ( x )⋅tg( x )
[ sec( x ) ] '=sec( x )⋅tg( x )R // .
18 . )Sea f ( x )=csc( x )Hallar [ f ( x ) ] ' .
[ sec( x ) ] '=limh→0
csc( x+h)−csc( x )h
[ csc( x ) ] '=limh→0
1sen ( x+h)
−1sen ( x )
h
[ csc( x ) ] '=limh→0
sen ( x )−sen( x+h )h⋅[ sen( x+h )⋅sen( x )]
[ csc( x ) ] '=limh→0
sen ( x )−sen( x )cos (h )−sen (h )cos( x )h⋅[ sen( x+h )⋅sen( x )]
[ csc( x ) ] '=limh→0
sen ( x )⋅[1−cos(h )]h⋅[ sen( x+h )⋅sen( x )]
−limh→0
sen (h)cos ( x )h⋅[ sen( x+h )⋅sen ( x )]
[ csc( x ) ] '=limh→0
sen ( x )⋅[1−cos(h )]h⋅[ sen( x+h )⋅sen( x )]
⋅[1+cos (h )][1+cos (h )]
−limh→0
sen(h )h⋅¿
⋅limh→ 0
cos ( x )[ sen( x+h )⋅sen ( x )]
¿
[ csc( x ) ] '=limh→0
[1−cos2( h)]h
⋅limh→ 0
sen (x )[cos(h )+1 ]⋅[sen ( x+h)⋅sen ( x )]
−(1 )⋅cos( x )sen( x+0 )⋅sen ( x )
[ csc( x ) ] '=limh→0
sen2 (h )h
⋅limh→0
sen( x )[cos(h )+1 ]⋅[sen (x+h )⋅sen( x ) ]
−cos(x )sen ( x )⋅sen( x )
[ csc( x ) ] '=limh→0
sen (h )h
⋅limh→0
sen (h )sen( x )[cos (h)+1 ]⋅[ sen( x+h )⋅sen ( x )]
−1sen ( x )
⋅cos ( x )sen( x )
[ csc( x ) ] '=(1 )⋅sen (0) sen( x )[cos(0 )+1 ]⋅[ sen( x+0 )⋅sen ( x )]
−csc( x )⋅ctg( x )
[ csc( x ) ] '=02 sen2 ( x )
−csc( x )⋅ctg( x )
[ csc( x ) ] '=−csc( x )⋅ctg( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
Si f (x) es una funcion diferenciable y f -1(x) su inversa, halle [f -1(x)]’
D V N Ing. ELECTICA POTENCIA Página 9
18 . )Sea f ( x )=csc( x )Hallar [ f ( x ) ] ' .
[ sec( x ) ] '=limh→0
csc( x+h)−csc( x )h
[ csc( x ) ] '=limh→0
1sen ( x+h)
−1sen ( x )
h
[ csc( x ) ] '=limh→0
sen ( x )−sen( x+h )h⋅[ sen( x+h )⋅sen( x )]
[ csc( x ) ] '=limh→0
sen ( x )−sen( x )cos (h )−sen (h )cos( x )h⋅[ sen( x+h )⋅sen( x )]
[ csc( x ) ] '=limh→0
sen ( x )⋅[1−cos(h )]h⋅[ sen( x+h )⋅sen( x )]
−limh→0
sen (h)cos ( x )h⋅[ sen( x+h )⋅sen ( x )]
[ csc( x ) ] '=limh→0
sen ( x )⋅[1−cos(h )]h⋅[ sen( x+h )⋅sen( x )]
⋅[1+cos (h )][1+cos (h )]
−limh→0
sen(h )h⋅¿
⋅limh→ 0
cos ( x )[ sen( x+h )⋅sen ( x )]
¿
[ csc( x ) ] '=limh→0
[1−cos2( h)]h
⋅limh→ 0
sen (x )[cos(h )+1 ]⋅[sen ( x+h)⋅sen ( x )]
−(1 )⋅cos( x )sen( x+0 )⋅sen ( x )
[ csc( x ) ] '=limh→0
sen2 (h )h
⋅limh→0
sen( x )[cos(h )+1 ]⋅[sen (x+h )⋅sen( x ) ]
−cos(x )sen ( x )⋅sen( x )
[ csc( x ) ] '=limh→0
sen (h )h
⋅limh→0
sen (h )sen( x )[cos (h)+1 ]⋅[ sen( x+h )⋅sen ( x )]
−1sen ( x )
⋅cos ( x )sen( x )
[ csc( x ) ] '=(1 )⋅sen (0) sen( x )[cos(0 )+1 ]⋅[ sen( x+0 )⋅sen ( x )]
−csc( x )⋅ctg( x )
[ csc( x ) ] '=02 sen2 ( x )
−csc( x )⋅ctg( x )
[ csc( x ) ] '=−csc( x )⋅ctg( x )R // .
19 .)Sea y=sen ( x )Hallar [ f −1 ( x ) ]' .[ y−1=arcsen (x )]≡[ sen( y )=x ]Por lo tanto( y−1 ) '=( x )' .
Dadoque( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=sen( y )∴( x ) '=cos( y )
sen( y )=cateto opuestohipotenusa
sen( y )=x1
∴ cos ( y )=√1−x2
Por lo tanto( y−1 )'=1
√1−x2R //,(−1< x<1 )
yaque sen ( y )es acot adaeneste int ervalo .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 10
20 .)Sea y=cos (x )Hallar [ f−1( x )] ' .[ y−1=arccos( x ) ]≡[cos ( y )=x ]Por lo tanto( y−1) '=( x ) ' .
Dado que( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=cos ( y )∴( x )'=−sen( y )
cos ( y )=catetoopuestohipotenusa
cos ( y )=x1
∴ sen ( y )=√1−x2
Por lo tanto( y−1 )'=−1
√1−x2R //,(−1< x<1 )
yaque cos( y )esacot ada eneste int ervalo .
21 .)Sea y=tg( x )Hallar [ f−1( x ) ] ' .[ y−1=arctg ( x )]≡ [tg ( y )=x ]Por lo tanto( y−1 ) '=( x )' .
Dado que( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=tg ( y )∴( x )'=sec2 ( y )
tg ( y )=catetoopuestocatetoadyacente
tg ( y )=x1
∴ cos( y )=1
√1+x2⇒sec ( y )=√1+x2
Por lo tanto( y−1 )'=1
1+x2R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 11
22 .)Sea y=ctg( x )Hallar [ f −1 ( x ) ]' .[ y−1=arcctg ( x )]≡ [ctg( y )=x ] Por lo tanto( y−1 )'=( x ) ' .
Dado que( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=ctg( y )∴( x ) '=−csc2( y )
ctg( y )=cateto adyacentecateto opuesto
ctg( y )=x1
∴sen ( y )=1
√1+x2⇒ csc ( y )=√1+x2
Por lo tanto( y−1 )'=−1
1+x2R // .
23 .)Sea y=sec( x )Hallar [ f −1 (x )] ' .[ y−1=arc sec( x ) ]≡[sec( y )=x ]Por lo tanto( y−1) '=( x ) ' .
Dadoque( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=sec( y )∴( x )'=sec( y )⋅tg( y )
sec( y )=hipotenusacatetoadyacente
sec( y )=x1
∴ tg( y )=√ x2−1
Por lo tanto( y−1 )'=1|x|⋅√ x2−1
R //, (1<|x|)
24 . )Sea y=csc ( x )Hallar[ f −1( x ) ] ' .[ y−1=arc csc ( x ) ]≡[csc ( y )=x ] Por lo tanto( y−1 )'=( x ) ' .
Dadoque( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=csc ( y )∴( x )'=−csc( y )⋅ctg( y )
csc ( y )=hipotenusacatetoopuesto
csc ( y )=x1
∴ ctg( y )=√x2−1
Por lo tanto( y−1 )'=−1|x|⋅√x2−1
R //, (1<|x|)
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
Demuestre formalmente las derivadas de las siguientes funciones
D V N Ing. ELECTICA POTENCIA Página 12
24 . )Sea y=csc ( x )Hallar[ f −1( x ) ] ' .[ y−1=arc csc ( x ) ]≡[csc ( y )=x ] Por lo tanto( y−1 )'=( x ) ' .
Dadoque( y−1 ) '=dydx
y ( x )'=dxdy
, entonces ( y−1) '=1
(dxdy )∴( y−1 )'=1
(x )'
x=csc ( y )∴( x )'=−csc( y )⋅ctg( y )
csc ( y )=hipotenusacatetoopuesto
csc ( y )=x1
∴ ctg( y )=√x2−1
Por lo tanto( y−1 )'=−1|x|⋅√x2−1
R //, (1<|x|)
31 .)[ arg senh( x ) ] '=1
√x2+1.
32 .)[ argcosh( x ) ]'=1
√x2−1,( x>1 ).
33 .)[ arg tgh( x ) ] '=11−x2
,(|x|<1)
34 . )[arg ctgh( x ) ]'=1
1−x2.(|x|>1 )
35 .)[ argsec h( x ) ] '=−1
x⋅√1−x2
( 0<x<1) .
36 .) [argcsc h( x ) ] '=−1|x|⋅√1+ x2
( x≠0) .
25 .)[ senh ( x ) ] '=cos( x ) .26 .) [cosh ( x ) ] '=senh ( x ).27 .) [ tgh( x ) ]'=sec2h( x ).28 .)[ ctgh( x ) ]'=−csc2 ( x ).29 .)[ sech ( x ) ]'=−sec h( x )⋅tgh( x ) .30 .)[ csc h( x ) ]'=−csc h( x )⋅ctgh( x ).
senh( x )=ex−e−x
2. cosh ( x )=e
x+e−x
2.
tgh( x )=ex−e− x
e x+e−x. ctgh( x )=ex+e− x
ex−e−x.
sec h( x )=2ex+e− x
. csch (x )=2e x−e− x
.
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 13
25 .) f (x )=senh( x )
[ senh( x ) ] '=limh→0
senh ( x+h)−senh (x )h
.
[ senh( x ) ] '=limh→0
ex+h−e−x−h
2−ex−e−x
2h
[ senh( x ) ] '=limh→0
ex+h−e−x−h−ex+e− x
2h
[ senh( x ) ] '=limh→0
(ex+h−ex )−(e−x−h−e− x )2h
[ senh( x ) ] '=limh→0
ex (eh−1 )2h
−limt→0
e−x (e−h−1 )2h
[ senh( x ) ] '=limh→0
ex
2⋅limh→0
eh−1h
−limh→0
e− x
2⋅limh→0
e−h−1h
[ senh( x ) ] '=ex
2⋅(1)−e
−x
2⋅(−1)
[ senh( x ) ] '=ex+e−x
2≡cosh ( x )R // .
26 .) f ( x )=cosh ( x )
[cosh ( x ) ] '=limh→0
cosh ( x+h)−cosh ( x )h
.
[cosh ( x ) ] '=limh→0
ex+h+e− x−h
2−e
x+e−x
2h
[cosh ( x ) ] '=limh→0
ex+h+e− x−h−ex−e− x
2h
[cosh ( x ) ] '=limh→0
(ex+h−ex )+(e− x−h−e−x )2h
[cosh ( x ) ] '=limh→0
ex (eh−1 )2h
+ limt→0
e−x (e−h−1 )2h
[cosh ( x ) ] '=limh→0
ex
2⋅limh→0
eh−1h
+ limh→0
e−x
2⋅limh→0
e−h−1h
[cosh ( x ) ] '=ex
2⋅(1)+e
−x
2⋅(−1)
[cosh ( x ) ] '=ex−e−x
2≡senh( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 14
27 .) f ( x )=tgh ( x )
[ tgh ( x ) ]'=limh→0
tgh ( x+h)−tgh( x )h
.
[ tgh ( x ) ]'=limh→0
e x+h−e− x−h
e x+h+e−x−h−e
x−e− x
e x+e−x
h
[ tgh ( x ) ]'=limh→0
(ex+h−e−x−h ) (ex+e−x )−(e x+h+e−x−h) (e x−e− x )h⋅(ex+h+e−x−h ) (ex+e−x )
[ tgh ( x ) ]'=limh→0
(e2 x+h−e−h+eh−e−2 x−h )− (e2 x+h+e−h−eh−e−2 x−h)h⋅(ex+h+e−x−h ) (ex+e−x )
[ tgh ( x ) ]'=limh→0
(e2 x+h−e2 x+h)+(e−2 x−h−e−2 x−h)+2 (eh−e−h )h⋅(ex+h+e−x−h ) (ex+e−x )
[ tgh ( x ) ]'=limh→0
2 [ (eh−1 )− (e−h−1 ) ]h⋅(ex+h+e−x−h ) (ex+e−x )
[ tgh ( x ) ]'=limh→0
2
(ex+h+e− x−h) (ex+e− x )⋅[ limh→0
(eh−1 )−(e−h−1 )h ]
[ tgh ( x ) ]'=2(ex+0+e− x−0) (ex+e− x )
⋅[ limh→0
eh−1h
−limh→0
e−h−1h ]
[ tgh ( x ) ]'=2
(ex+e− x )2⋅(1−(−1 ))=[2(ex+e− x ) ]
2
=sec h2 (x )R // .
28 .) f (x )=ctgh( x )
[ctgh( x ) ] '=limh→ 0
ctgh(x+h )−ctgh( x )h
.
[ctgh( x ) ] '=limh→ 0
ex+h+e− x−h
ex+h−e−x−h−ex+e− x
ex−e−x
h
[ctgh( x ) ] '=limh→ 0
(ex+h+e−x−h ) (ex−e−x )−(ex+h−e−x−h ) (ex+e−x )h⋅(e x+h+e−x−h) (e x+e− x )
[ctgh( x ) ] '=limh→ 0
(e2 x+h+e−h−eh−e−2 x−h)−(e2 x+h−e−h+eh−e−2 x−h )h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
(e2 x+h−e2 x+h)+(e−2 x−h−e−2 x−h )−2 (eh−e−h)h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
−2 [ (eh−1 )−(e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
−2(ex+h−e−x−h) (e x−e− x )
⋅[ limh→0 (eh−1 )−(e−h−1 )h ]
[ctgh( x ) ] '=−2(ex+0−e−x−0) (e x−e− x )
⋅[ limh→0 eh−1h
−limh→0
e−h−1h ]
[ctgh( x ) ] '=−2
(ex−e−x )2⋅(1−(−1 ) )
[ctgh( x ) ] '=−[2(e x−e− x ) ]2
[ctgh( x ) ] '=−csch2( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 15
28 .) f (x )=ctgh( x )
[ctgh( x ) ] '=limh→ 0
ctgh(x+h )−ctgh( x )h
.
[ctgh( x ) ] '=limh→ 0
ex+h+e− x−h
ex+h−e−x−h−ex+e− x
ex−e−x
h
[ctgh( x ) ] '=limh→ 0
(ex+h+e−x−h ) (ex−e−x )−(ex+h−e−x−h ) (ex+e−x )h⋅(e x+h+e−x−h) (e x+e− x )
[ctgh( x ) ] '=limh→ 0
(e2 x+h+e−h−eh−e−2 x−h)−(e2 x+h−e−h+eh−e−2 x−h )h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
(e2 x+h−e2 x+h)+(e−2 x−h−e−2 x−h )−2 (eh−e−h)h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
−2 [ (eh−1 )−(e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[ctgh( x ) ] '=limh→ 0
−2(ex+h−e−x−h) (e x−e− x )
⋅[ limh→0 (eh−1 )−(e−h−1 )h ]
[ctgh( x ) ] '=−2(ex+0−e−x−0) (e x−e− x )
⋅[ limh→0 eh−1h
−limh→0
e−h−1h ]
[ctgh( x ) ] '=−2
(ex−e−x )2⋅(1−(−1 ) )
[ctgh( x ) ] '=−[2(e x−e− x ) ]2
[ctgh( x ) ] '=−csch2( x )R // .
29 .) f (x )=sec h( x )
[sec h( x ) ] '=limh→0
sec h( x+h )−sec h( x )h
.
[sec h( x ) ] '=limh→0
2
ex+h+e− x−h−2ex+e−x
h
[sec h( x ) ] '=limh→0
2 (ex+e− x )−2 (e x+h+e−x−h)h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
2 (−ex+h+ex−e− x−h+e−x )h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2 [ (ex+h−ex )+(e−x−h−e−x ) ]h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2 [ ex (eh−1 )+e−x (e−h−1 ) ]h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2(ex+h+e− x−h ) (ex+e− x )
⋅[ limh→0 ex (eh−1 )+e−x (e−h−1 )h ]
[sec h( x ) ] '=−2(ex+0+e− x−0 ) (ex+e− x )
⋅[ex limh→0 eh−1h
+e− x limh→0
e−h−1h ]
[sec h( x ) ] '=−2(ex+e−x )2
⋅(ex−e−x )
[sec h( x ) ] '=−2(ex+e−x )
⋅(ex−e− x )(ex+e−x )
[sec h( x ) ] '=−sech( x )⋅tgh( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 16
29 .) f (x )=sec h( x )
[sec h( x ) ] '=limh→0
sec h( x+h )−sec h( x )h
.
[sec h( x ) ] '=limh→0
2
ex+h+e− x−h−2ex+e−x
h
[sec h( x ) ] '=limh→0
2 (ex+e− x )−2 (e x+h+e−x−h)h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
2 (−ex+h+ex−e− x−h+e−x )h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2 [ (ex+h−ex )+(e−x−h−e−x ) ]h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2 [ ex (eh−1 )+e−x (e−h−1 ) ]h⋅(e x+h+e−x−h) (e x+e−x )
[sec h( x ) ] '=limh→0
−2(ex+h+e− x−h ) (ex+e− x )
⋅[ limh→0 ex (eh−1 )+e−x (e−h−1 )h ]
[sec h( x ) ] '=−2(ex+0+e− x−0 ) (ex+e− x )
⋅[ex limh→0 eh−1h
+e− x limh→0
e−h−1h ]
[sec h( x ) ] '=−2(ex+e−x )2
⋅(ex−e−x )
[sec h( x ) ] '=−2(ex+e−x )
⋅(ex−e− x )(ex+e−x )
[sec h( x ) ] '=−sech( x )⋅tgh( x )R // .
30 .) f (x )=csc h( x )
[csc h( x ) ] '=limh→0
csch( x+h )−csc h( x )h
.
[csc h( x ) ] '=limh→0
2
ex+h−e−x−h−2ex−e−x
h
[csc h( x ) ] '=limh→0
2 (ex−e−x )−2 (ex+h−e−x−h)h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
2 (−ex+h+e x+e−x−h−e−x )h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2 [ (ex+h−ex )−(e− x−h−e−x ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2 [ ex (eh−1 )−e− x (e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2(ex+h−e−x−h) (ex−e− x )
⋅[ limh→0
ex (eh−1 )−e−x (e−h−1 )h ]
[csc h( x ) ] '=−2(ex+0−e−x−0 ) (ex−e− x )
⋅[ex limh→0 eh−1h
−e−x limh→0
e−h−1h ]
[csc h( x ) ] '=−2(ex−e−x )2
⋅(ex+e−x )
[csc h( x ) ] '=−2(ex−e−x )
⋅(ex+e− x )(ex−e−x )
[csc h( x ) ] '=−csch( x )⋅ctgh( x )R // .
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
Si f (x) es una funcion diferenciable y f -1(x) su inversa, halle [f -1(x)]’
D V N Ing. ELECTICA POTENCIA Página 17
30 .) f (x )=csc h( x )
[csc h( x ) ] '=limh→0
csch( x+h )−csc h( x )h
.
[csc h( x ) ] '=limh→0
2
ex+h−e−x−h−2ex−e−x
h
[csc h( x ) ] '=limh→0
2 (ex−e−x )−2 (ex+h−e−x−h)h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
2 (−ex+h+e x+e−x−h−e−x )h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2 [ (ex+h−ex )−(e− x−h−e−x ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2 [ ex (eh−1 )−e− x (e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )
[csc h( x ) ] '=limh→0
−2(ex+h−e−x−h) (ex−e− x )
⋅[ limh→0
ex (eh−1 )−e−x (e−h−1 )h ]
[csc h( x ) ] '=−2(ex+0−e−x−0 ) (ex−e− x )
⋅[ex limh→0 eh−1h
−e−x limh→0
e−h−1h ]
[csc h( x ) ] '=−2(ex−e−x )2
⋅(ex+e−x )
[csc h( x ) ] '=−2(ex−e−x )
⋅(ex+e− x )(ex−e−x )
[csc h( x ) ] '=−csch( x )⋅ctgh( x )R // .
31 .)Sea y=senh ( x )Hallar [ f−1 (x )] ' .
x=senh( y )
sen( y )=ey−e− y
2
x=ey−e− y
22 x=e y−e− y
e y (2 x )=e y (e y−e− y )2 xe y=e2 y−1 ; t=e y
e2 y−2xe y−1=0t2−2xt−1=0
t=2 x±√4 x2+42
t=x+√ x2+1∨t=x−√x2+1e y=x+√x2+1y−1=ln (x+√ x2+1 )
( y−1) '=1
x+√x2+1⋅(1+x√x2+1 )
( y−1) '=1x+√x2+1
⋅(x+√ x2+1√x2+1 )
( y−1) '=1
√x2+1Por lo tanto( y−1 )'=1
√ x2+1R //
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 18
32 .)Sea y=cosh ( x )Hallar [ f−1 (x )] ' .
x=cosh ( y )
cos ( y )=e y+e− y
2
x=ey+e− y
22 x=e y+e− y
e y (2 x )=e y (e y+e− y )2 xe y=e2 y+1; t=e y
e2 y−2xe y+1=0t2−2xt+1=0
t=2 x±√4 x2−42
t=x+√ x2−1∨t=x−√x2−1e y=x+√x2−1y−1=ln (x+√ x2−1 )
( y−1) '=1
x+√x2−1⋅(1+ x√x2−1 )
( y−1) '=1x+√x2−1
⋅( x+√x2−1√x2−1 )
( y−1) '=1
√x2−1Por lo tanto( y−1 )'=1
√ x2−1R //,( x>1)
33 .)Sea y=tgh( x )Hallar [ f−1( x ) ] ' .
x=tgh( y )
tg( y )=ey−e− y
e y+e− y
x=ey−e− y
e y+e− y
x (e y+e− y )=e y−e− y
xe y (e y+e− y )=e y (e y−e− y )xe2 y+ x=e2 y−1e2 y−xe 2 y=x+1e2 y (1−x )=x+1
e2 y=x+11−x
2 y−1=ln(x+11−x )y−1=1
2ln(x+11−x )
( y−1) '=12⋅(1−xx+1 )⋅1−x+1+x
(1−x )2
( y−1) '=12⋅2
(1+x ) (1−x )
Por lo tanto( y−1 )'=11−x2
R //,(|x|<1)
34 . )Sea y=ctgh( x )Hallar [ f −1( x ) ] ' .
x=ctgh( y )
ctg( y )=ey+e− y
e y−e− y
x=ey+e− y
e y−e− y
x (e y−e− y )=e y+e− y
xe y (e y−e− y )=e y (e y+e− y )xe2 y−x=e2 y+1xe2 y−e2 y=x+1e2 y ( x−1 )=x+1
e2 y=x+1x−1
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DERIVADA DE UNA FUNCIÓN POR MEDIO DE LÍMITES
D V N Ing. ELECTICA POTENCIA Página 19
34 . )Sea y=ctgh( x )Hallar [ f −1( x ) ] ' .
x=ctgh( y )
ctg( y )=ey+e− y
e y−e− y
x=ey+e− y
e y−e− y
x (e y−e− y )=e y+e− y
xe y (e y−e− y )=e y (e y+e− y )xe2 y−x=e2 y+1xe2 y−e2 y=x+1e2 y ( x−1 )=x+1
e2 y=x+1x−1
2 y−1=ln(x+1x−1 )y−1=1
2ln(x+1x−1 )
( y−1) '=12⋅( x−1x+1 )⋅x−1−x−1
( x−1 )2
( y−1) '=12⋅−2
( x+1 ) (x−1 )
( y−1) '=−1x2−1
Por lo tanto( y−1)'=11−x2
R //,(|x|>1)
35 .)Sea y=sech ( x )Hallar [ f−1 (x )] ' .
x=sec h( y )
sec( y )=2e y+e− y
x=2e y+e− y
x (e y+e− y )=2xe y (e y+e− y )=2e yxe2 y+ x=2e y ; t=e y
xt2−2 t+ x=0
t=2±√4−4 x22 x
t=1+√1−x2
x∨t=
1−√1−x2
x
e y=1+√1−x2
x
y−1=ln(1+√1−x2
x )
( y−1) '=x
1+√1−x2⋅(−x2
√1−x2−1−√1−x2
x2)
( y−1) '=x
1+√1−x2⋅(−x2−√1−x2−1+x2
x2√1−x2 )( y−1) '=−x
1+√1−x2⋅(1+√1−x2
x2√1−x2 )Por lo tanto( y−1 )'=−1
x √1−x2R //,(0< x<1 )
36 .) Sea y=csc h( x )Hallar [ f −1 ( x ) ]' .
x=csc h( y )
csc ( y )=2e y−e− y
x=2e y−e− y
x (e y−e− y )=2xe y (e y−e− y )=2e yxe2 y−x=2e y ; t=e y
xt2−2 t−x=0
t=2±√4+4 x22 x
t=1+√1+x2x
∨t=1−√1+ x2x
e y=1+√1+x2x
y−1=ln(1+√1+x2x )
( y−1) '=x
1+√1+x2⋅(x
2
√1+x2−1−√1+x2
x2)
( y−1) '=x
1+√1+x2⋅(x2−√1+x2−1−x2
x2 √1+x2 )( y−1) '=−x
1+√1+x2⋅(1+√1+x2x2 √1+x2 )
Por lo tanto( y−1 )'=−1|x|⋅√1+x2
R //,( x≠0 )