demostraciones de derivadas por medio de limites

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 //



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 )


11tln ( t+1)


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 )



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 // .



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 // .



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 // .


Derivadas de funciones trigonométricas y sus inversas.


13 . )Sea f ( x )=sen ( x )Hallar [ f ( x )] ' .

[ sen (x )] '=limh→0

sen( x+h )−sen ( x )h


sen( x )cos (h)+sen(h )cos( x )−sen ( x )h


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 // .



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 // .



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 .


[ 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


cos ( x+h)sen( x+h )

−cos ( x )sen( x )

h


sen( x )cos ( x+h)−sen (x+h )cos( x )h⋅[sen ( x+h)⋅sen( x ) ]

;α=x , β=x+h.



sen( x−h−x )h⋅[sen ( x+h)⋅sen( x ) ]


−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


⋅[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 ) ] '=−csc( x )⋅ctg( x )R // .


Si f (x) es una funcion diferenciable y f -1(x) su inversa, halle [f -1(x)]’


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


−limh→0

sen (h)cos ( x )h⋅[ sen( x+h )⋅sen ( x )]

[ csc( x ) ] '=limh→0


⋅[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 ) ] '=02 sen2 ( 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 .



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


(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 )' .


y ( x )'=dxdy


(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 // .



22 .)Sea y=ctg( x )Hallar [ f −1 ( x ) ]' .[ y−1=arcctg ( x )]≡ [ctg( y )=x ] Por lo tanto( y−1 )'=( x ) ' .


y ( x )'=dxdy


(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 ) ' .


y ( x )'=dxdy


(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 ) ' .


y ( x )'=dxdy


(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|)


Demuestre formalmente las derivadas de las siguientes funciones


24 . )Sea y=csc ( x )Hallar[ f −1( x ) ] ' .[ y−1=arc csc ( x ) ]≡[csc ( y )=x ] Por lo tanto( y−1 )'=( x ) ' .


y ( x )'=dxdy


(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

.



25 .) f (x )=senh( x )

[ senh( x ) ] '=limh→0

senh ( x+h)−senh (x )h

.


ex+h−e−x−h

2−ex−e−x

2h


ex+h−e−x−h−ex+e− x

2h


(ex+h−ex )−(e−x−h−e− x )2h


ex (eh−1 )2h

−limt→0

e−x (e−h−1 )2h


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

.


ex+h+e− x−h

2−e

x+e−x

2h


ex+h+e− x−h−ex−e− x

2h


(ex+h−ex )+(e− x−h−e−x )2h


ex (eh−1 )2h

+ limt→0

e−x (e−h−1 )2h


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 // .



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

.


ex+h+e− x−h


ex−e−x

h


(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 )


(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 )


(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 )


−2 [ (eh−1 )−(e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−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 // .



28 .) f (x )=ctgh( x )


ctgh(x+h )−ctgh( x )h

.


ex+h+e− x−h


ex−e−x

h


(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 )


(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 )


(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 )


−2 [ (eh−1 )−(e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−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

.


2

ex+h+e− x−h−2ex+e−x

h


2 (ex+e− x )−2 (e x+h+e−x−h)h⋅(e x+h+e−x−h) (e x+e−x )


2 (−ex+h+ex−e− x−h+e−x )h⋅(e x+h+e−x−h) (e x+e−x )


−2 [ (ex+h−ex )+(e−x−h−e−x ) ]h⋅(e x+h+e−x−h) (e x+e−x )


−2 [ ex (eh−1 )+e−x (e−h−1 ) ]h⋅(e x+h+e−x−h) (e x+e−x )


−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 // .



29 .) f (x )=sec h( x )


sec h( x+h )−sec h( x )h

.


2

ex+h+e− x−h−2ex+e−x

h


2 (ex+e− x )−2 (e x+h+e−x−h)h⋅(e x+h+e−x−h) (e x+e−x )


2 (−ex+h+ex−e− x−h+e−x )h⋅(e x+h+e−x−h) (e x+e−x )


−2 [ (ex+h−ex )+(e−x−h−e−x ) ]h⋅(e x+h+e−x−h) (e x+e−x )


−2 [ ex (eh−1 )+e−x (e−h−1 ) ]h⋅(e x+h+e−x−h) (e x+e−x )


−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 )


+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

.


2

ex+h−e−x−h−2ex−e−x

h


2 (ex−e−x )−2 (ex+h−e−x−h)h⋅(e x+h−e− x−h ) (ex−e−x )


2 (−ex+h+e x+e−x−h−e−x )h⋅(e x+h−e− x−h ) (ex−e−x )


−2 [ (ex+h−ex )−(e− x−h−e−x ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−2 [ ex (eh−1 )−e− x (e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−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 )


−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 // .


Si f (x) es una funcion diferenciable y f -1(x) su inversa, halle [f -1(x)]’


30 .) f (x )=csc h( x )


csch( x+h )−csc h( x )h

.


2

ex+h−e−x−h−2ex−e−x

h


2 (ex−e−x )−2 (ex+h−e−x−h)h⋅(e x+h−e− x−h ) (ex−e−x )


2 (−ex+h+e x+e−x−h−e−x )h⋅(e x+h−e− x−h ) (ex−e−x )


−2 [ (ex+h−ex )−(e− x−h−e−x ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−2 [ ex (eh−1 )−e− x (e−h−1 ) ]h⋅(e x+h−e− x−h ) (ex−e−x )


−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 )


−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 //



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



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 )

demostraciones de derivadas por medio de limites

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