School Work > Essays & Theses">
Derivadas de Las Funciones Trigonometricas
Derivadas de Las Funciones Trigonometricas
Derivadas de Las Funciones Trigonometricas
+
| |
=
|
\ .
Para calcular este lmite se utilizan los siguientes conceptos previamente estudiados:
0
( )
1
h
Sen h
Lim
h
=
0
( ) 1
0
h
Cos h
Lim
h
=
) ( ) ( ) ( ) ( ) ( h Sen x Cos h Cos x Sen h x Sen + = +
Por lo tanto desarrollando el lmite se tiene:
|
.
|
\
|
+
=
h
x Sen h x Sen
Lim
dx
dy
h
) ( ) (
0
Definicin de derivada
|
.
|
\
|
+
=
h
x Sen h Sen x Cos h Cos x Sen
Lim
dx
dy
h
) ( ) ( ) ( ) ( ) (
0
Aplicando suma de arcos
|
.
|
\
|
+
=
h
h Sen x Cos h Cos x Sen
Lim
dx
dy
h
) ( ) ( ) 1 ) ( )( (
0
Factorizando el numerador
|
.
|
\
|
+
|
.
|
\
|
=
h
h Sen
x Cos Lim
h
h Cos
x Sen Lim
dx
dy
h h
) (
) (
1 ) (
) (
0 0
Suma de Limites
|
.
|
\
|
+
|
.
|
\
|
=
h
h Sen
Lim x Cos
h
h Cos
Lim x Sen
dx
dy
h h
) (
) (
1 ) (
) (
0 0
Sacando las constantes fuera
del lmite
) ( 1 ) ( 0 ) ( x Cos x Cos x Sen
dx
dy
= + =
Por los lmites conocidos
De donde: ( ) ( )
d
Sen x Cos x
dx
=
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
Si u es una funcin diferenciable de x, es posible aplicar la regla de la cadena as:
dx
du
du
dy
dx
dy
=
en donde u Sen y = para obtener como resultado:
( )
d du
Sen u Cos u
dx dx
=
Ejemplos:
1.
( ) ( )
2 2 2 2 2
d d
Sen x Cos x x Cos x
dx dx
= =
2.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2
2
d d d
x Sen x x Sen x Sen x x x Cos x x Sen x
dx dx dx
= + = +
3.
( )
( ) ( ) ( ) ( )
( ) ( )
2 2
d d
x Sen x Sen x x
Sen x xCos x Sen x
d
dx dx
dx x x x
| |
= =
|
\ .
4.
( )
2
2
Sen x
d
dx x
| |
=
|
\ .
( ) ( ) ( ) ( )
( )
2 2 2 2
2
2
d d
x Sen x Sen x x
dx dx
x
=
( ) ( ) ( ) ( ) ( )
2 2
4
2 2 x Sen x Cos x x Sen x
x
= =
( ) ( ) ( )
2 2
4
2 2 x Sen x Cos x xSen x
x
= =
2 x
=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
3 4
2 xSen x Cos x Sen x xSen x Cos x Sen x
x x
=
Derivada de
( )
y Cos u =
Para obtener esta derivada hay que tener presente las siguientes identidades:
w
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
|
.
|
\
|
=
|
.
|
\
|
= u Cos u Sen u sen u Cos
2 2
Luego:
dx
du
u Sen
dx
du
u Sen
u
dx
d
u u sen
dx
d
u Cos
dx
d
=
|
.
|
\
|
=
=
|
.
|
\
|
|
.
|
\
|
=
|
.
|
\
|
=
2 2
cos
2
De donde se puede concluir:
( )
dx
du
u Sen u Cos
dx
d
=
Ejemplos:
1.
( ) ( )
3 3 3 3 3
d d
Cos x Sen x x Sen x
dx dx
= =
2.
( ) ( ) ( ) ( ) ( ) ( )
4 4 3
2 3 2 3 8 3
d d d
x Cos x x Cos x x Sen x
dx dx dx
+ = + =
3.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2
2
d d d
x Cos x x Cos x Cos x x x Sen x xCos x
dx dx dx
= + = +
4.
( )
( )
1
Sen x
d
dx Cos x
| |
|
|
\ .
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2
1 1
1
d d
Cos x Sen x Sen x Cos x
dx dx
Cos x
= =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2
1
1
Cos x Cos x Sen x Sen x
Cos x
= =
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
2 2
2 2
1
1 1
Cos x
Cos x Cos x Sen x
Cos x Cos x
= = =
( ) ( ) ( )
1 1 Cos x
=
( ) ( )
1 Cos x
( ) ( )
( ) ( )
1
1
1
Cos x
Cos x
=
Derivada de
( )
y Tan x =
w
w
w
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
( )
|
|
.
|
\
|
=
x Cos
x Sen
dx
d
x Tan
dx
d
Definicin funcin tangente
( )
( ) ( )
x Cos
x Cos
dx
d
x Sen x Sen
dx
d
x Cos
x Tan
dx
d
2
=
Derivada de un cociente
( )
x Cos
x Sen x Sen x Cos x Cos
x Tan
dx
d
2
= Resolviendo la derivada
( ) x Sec
x Cos x Cos
x Sen x Cos
x Tan
dx
d
2
2 2
2 2
1
= =
=
Agrupando trminos
De manera que si u es una funcin diferenciable de x, aplicando la regla de la cadena a
la funcin u y tan = se puede concluir:
dx
du
u Sec u Tan
dx
d
2
=
Ejemplos :
1.
( ) ( ) ( )
2 2
5 5 5 5 5
d d
Tan x Sec x x Sec x
dx dx
| |
= =
|
\ .
2.
( ) ( ) ( ) ( ) ( ) ( )
2 2 2
d d d
x Tan x x Tan x Tan x x
dx dx dx
= + =
( ) ( ) ( ) ( ) ( ) ( )
2 2 2
2 2 x Sec x Tan x x x xSec x Tan x + = +
3.
( ) ( ) ( ) ( ) ( ) ( )
2
1
d d d
x Tan x x Tan x Sec x
dx dx dx
+ = + = +
4.
( ) ( ) ( ) ( ) ( )
( )
2 2 2
1 1 1
d d d
x Tan x Tan Tan x
x x x
dx dx dx
= + =
( ) ( ) ( )
( )
2 2
1 1 1
2
d
x Sec Tan x
x x x
dx
(
(
+ =
(
2
x
( )
2
1
x
( )
( )
( )
2
1 1
2 Sec x Tan
x x
| |
+ =
|
|
\ .
( )
( )
( )
2
1 1
2 Sec x Tan
x x
+
w
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
5.
( ) ( ) ( ) ( )
2
1 1 1
d d
Tan x Sec x x
dx dx
+ = + + =
( ) ( ) ( ) ( )
2 2
1 1 Sec x Sec x ( + = +
Derivada de ( )
y Cot u =
|
|
.
|
\
|
=
x Sen
x Cos
dx
d
x Cot
dx
d
Definicin de Cotangente
( ) ( )
x Sen
x Sen
dx
d
x Cos x Cos
dx
d
x Sen
x Cot
dx
d
2
=
Derivada de un cociente
x Sen
x Cos x Cos x Sen x Sen
x Cot
dx
d
2
=
Resolviendo la derivada
( ) x Csc
x Sen x Sen
x Cos x Sen
x Cot
dx
d
2
2 2
2 2
1
=
=
=
Factorizando y Simplifando
De manera que si u es una funcin diferenciable de x, aplicando la regla de la cadena a
la funcin u Cot y = se puede concluir:
dx
du
u Csc u Cot
dx
d
2
=
Ejemplos :
1.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
2 2
d d
Cot x Cot x Cot x Cot x Csc x
dx dx
= = =
( ) ( )
2
2Cot x Csc x
2.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
5 2 5 5 2 5 5 5
d d d
Cot x Cot x Cot x Cot x Csc x x
dx dx dx
| |
= = =
|
\ .
( ) ( ) ( ) ( )( ) ( ) ( )
2 2
2 5 5 5 10 5 5 Cot x Csc x Cot x Csc x ( = (
3.
( )
( ) ( )
2
2
3 1
1
3
d
Ctg x
dx
d
dx Ctg x
| |
| =
|
\ .
( ) ( ) ( )
( ) ( )
0
2
2
2
1 3
3
d
Ctg x
dx
Ctg x
| |
|
|
\ .
=
w
w
w
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
( ) ( ) ( )
( )
( ) ( )
( )
( )
( ) ( )
( )
2 2 2
2 2 2 2
2 2 2 2 2 2
3 3
3 6 6 3
3 3 3
d
Csc x x
Csc x x x Csc x
dx
Ctg x Ctg x Ctg x
= =
4.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
1 1
d d d
x Ctg x x Ctg x Csc x Csc x
dx dx dx
+ = + = + =
5.
( ) ( ) ( ) ( ) ( )
( )
2 2 2
1 1 1
d d d
x Ctg x Ctg Ctg x
x x x
dx dx dx
= + =
( ) ( ) ( ) ( )
( )
2 2 2
1 1 1
d d
x Csc Ctg x
x x x
dx dx
+ =
2
x
( ) ( )
2
2
1 1
Csc
x
x
( )
( )
1
2 Ctg x
x
| |
+ =
|
\ .
( )
( )
( )
2
1 1
2 Csc x Ctg
x x
+
Derivada de ( )
y Sec u =
( ) ( )
( )
1 d d
Sec x
dx dx Cos x
| |
=
|
|
\ .
Definicin de Secante
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
2
1 1
d d
Cos x Cos x
d
dx dx
Sec x
dx Cos x
=
Derivada de un cociente
( ) ( )
( ) ( )
( )
2
1 Sen x
d
Sec x
dx Cos x
=
Resolviendo la derivada
( ) ( )
( )
( )
( )
( ) ( )
( ) ( )
2
1
Sen x Sen x
d
Sec x Tan x Sec x
dx Cos x Cos x Cos x
= = =
Simplificando y factorizado
De manera que si u es una funcin diferenciable de x, aplicando la regla de la cadena a
la funcin u Sec y = se puede concluir:
( ) ( ) ( )
d du
Sec u Tan u Sec u
dx dx
=
Ejemplos :
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
1.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2
2 2
d d
Sec x Sec x Sec x Sec x Sec x Tan x
dx dx
= = =
( ) ( )
2
2Sec x Tan x
2.
( ) ( ) ( ) ( ) ( ) ( )
2
5 2 5 5
d d
Sec x Sec x Sec x
dx dx
= =
( ) ( ) ( ) ( ) ( )
2 5 5 5 5
d
Sec x Sec x Tan x x
dx
| |
=
|
\ .
( ) ( ) ( ) ( ) ( )( )
2 5 5 5 5 Sec x Sec x Tan x =
( ) ( ) ( ) ( )
2
10 5 5 Sec x Tan x
3.
( )
( ) ( )
2
2
1
1
d
Sec x
dx
d
dx Sec x
| |
| =
|
\ .
( ) ( ) ( )
( ) ( )
0
2
2
2
1
d
Sec x
dx
Sec x
| |
|
|
\ .
=
( ) ( ) ( ) ( )
( ) ( )
2 2 2
2
2
d
Sec x Tan x x
dx
Sec x
=
( ) ( ) ( )
( )
( ) ( )
2 2
2
2
2 Sec x Tan x x
Sec x
=
( ) ( )
2
2x Sec x
( )
( )
( ) ( )
2
2
2
Tan x
Sec x
=
( ) ( )
( )
2
2
2x Tan x
Sec x
=
4.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
2
d d d
x Sec x x Sec x x Sec x Tan x
dx dx dx
+ = + = +
5.
( ) ( ) ( ) ( ) ( ) ( )
2 2 2
3 3 3
d d d
x Sec x x Sec x Sec x x
dx dx dx
= + =
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
( ) ( ) ( ) ( ) ( )( )
2
3 3 3 3 2
d
x Sec x Tan x x Sec x x
dx
+ =
( ) ( ) ( )( ) ( ) ( )
2
3 3 3 2 3 x Sec x Tan x x Sec x + =
( ) ( ) ( ) ( ) ( )
2
3 3 3 2 3 x Sec x Tan x x Sec x +
Derivada de ( )
y Csc u =
( )
( )
1 d d
Csc x
dx dx Sen x
| |
=
|
|
\ .
Definicin de la Cosecante
( )
( ) ( ) ( ) ( )
( )
2
1 1
d d
Sen x Sen x
d
dx dx
Csc x
dx Sen x
=
Derivada de un Cociente
( )
( ) ( )
( )
2
1 Cos x
d
Csc x
dx Sen x
=
Resolviendo la derivada.
( )
( )
( )
( )
( ) ( )
2
1
Cos x Cos x
d
Csc x
dx Sen x Sen x Sen x
= = =
( ) ( ) ( )
d
Csc x Cot x Csc x
dx
=
Simplificando y factorizando
De manera que si u es una funcin diferenciable de x, aplicando la regla de la cadena a
la funcin ( )
y Csc u = se puede concluir:
( ) ( ) ( )
d du
Csc u Cot u Csc u
dx dx
=
Ejemplos :
1.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2
2 2
d d
Csc x Csc x Csc x Csc x Csc x Ctg x
dx dx
= = =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2
2 2 Csc x Csc x Ctg x Csc x Ctg x =
2.
( ) ( ) ( ) ( ) ( ) ( )
2 3 3 3
2
d d
Csc x Csc x Csc x
dx dx
= =
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
( ) ( ) ( ) ( ) ( ) ( )
3 3 3 3
2
d
Csc x Csc x Ctg x x
dx
=
( ) ( ) ( ) ( ) ( )( )
3 3 3 2
2 3 Csc x Csc x Ctg x x =
( ) ( ) ( ) ( )
( )
( )
( )
3
2 2 3 3 2
2 3 3
1
6 6
Cos x
x Csc x Ctg x x
Sen x Sen x
=
( )
( )
3
2
3 3
6
Cos x
x
Sen x
3.
( )
( ) ( )
1
1
d
Csc x
dx
d
dx Csc x
| |
=
|
|
\ .
( ) ( ) ( )
( ) ( )
0
2
1
d
Csc x
dx
Csc x
| |
|
|
\ .
=
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
2
1
1
Csc x
Csc x Ctg x
Csc x
=
( )
( )
( ) ( )
2
Ctg x
Csc x
( ) ( )
( ) ( )
( )
( )
Ctg x
Cos x
Csc x
Sen x
=
( )
Sen x
( )
1
Cos x =
4.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
d d d
x Csc x x Csc x x Csc x Ctg x
dx dx dx
+ = + = +
5.
( ) ( ) ( ) ( ) ( ) ( )
2 2 2
3 3 3
d d d
x Csc x x Csc x Csc x x
dx dx dx
= + =
( ) ( ) ( ) ( ) ( )( )
2
3 3 3 3 2
d
x Csc x Ctg x x Csc x x
dx
+ =
( ) ( ) ( ) ( ) ( )( )
2
3 3 3 3 2 x Csc x Ctg x Csc x x + =
( ) ( ) ( ) ( ) ( ) ( )
2
3 3 3 2 3 x Csc x Ctg x x Csc x + =
Ejercicios Propuestos :
Encontrar la derivada de las siguientes expresiones:
1.
( )
3 y x Sen x = 2.
( ) ( )
2 y Cos x Tan x =
3.
( )
3
y t Cos t = 4.
( ) ( )
4 y Sec t Tan t = +
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CALCULO DIFERENCIAL
Escuela Colombiana de Ingeniera
4.- Derivadas Funciones Trigonomtricas
5.
( )
Tan x
y
x
=
6.
( )
( )
1
sen x
y
Cos x
=
+
7.
( ) ( )
x
y
Sen x Cos x
=
+
8.
( )
( )
1 Tan x
y
Sec x
=
9.
( )
2
Sen x
y
x
= 10.
( )
2
Sen x
y
x
=
11.
( ) ( ) ( ) ( )
y x Cos x Sen x = 12.
( )
3
2 y Sen x =
13.
( ) ( )
2 2
y Sen x Cos x = + 14.
( )
2
1 x
y
xSen x
+
=
15.
( )
4 2
3 y sen x x = + 16.
( )
2
2 y xSen x =
17. ( ) ( )
3
y Sen Cos t = 18.
( ) ( )
2
y Sen Cos x =
19. ( )
2 5
2 y x Sen x = 20. ( ) ( )
y Sec x Sen x =
w
w
w
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