Función
Derivada
s
e
n
(
x
)
{\displaystyle \,sen(x)}
c
o
s
(
x
)
{\displaystyle \,cos(x)}
c
o
s
(
x
)
{\displaystyle \,cos(x)}
−
s
e
n
(
x
)
{\displaystyle -\,sen(x)}
t
a
n
(
x
)
{\displaystyle \,tan(x)}
s
e
c
2
(
x
)
{\displaystyle \,sec^{2}(x)}
c
o
t
(
x
)
{\displaystyle \,cot(x)}
−
c
s
c
2
(
x
)
{\displaystyle -\,csc^{2}(x)}
s
e
c
(
x
)
{\displaystyle \,sec(x)}
s
e
c
(
x
)
tan
(
x
)
{\displaystyle \,sec(x)\tan(x)}
c
s
c
(
x
)
{\displaystyle \,csc(x)}
−
c
s
c
(
x
)
cot
(
x
)
{\displaystyle -\,csc(x)\cot(x)}
a
r
c
s
e
n
(
x
)
{\displaystyle \,arcsen(x)}
1
1
−
x
2
{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}
a
r
c
c
o
s
(
x
)
{\displaystyle \,arccos(x)}
−
1
1
−
x
2
{\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}
a
r
c
t
a
n
(
x
)
{\displaystyle \,arctan(x)}
1
x
2
+
1
{\displaystyle {\frac {1}{x^{2}+1}}}
La derivación de las funciones trigonométricas es el proceso matemático de encontrar el ritmo al cual una función trigonométrica cambia respecto de la variable independiente; es decir, la derivada de la función. Las funciones trigonométricas más habituales son las funciones sen(x) , cos(x) y tan(x) . Por ejemplo, al derivar f(x) = sen(x) , se está calculando la función f'(x) tal que da el ritmo de cambio del sen(x) en cada punto x .
f
′
(
x
)
=
cos
(
x
)
{\displaystyle f'(x)=\cos(x)\,}
Derivada de la función coseno
editar
Dada la función
f
(
x
)
=
cos
(
x
)
=
sen
(
x
+
π
2
)
{\displaystyle f(x)=\cos(x)=\operatorname {sen}(x+{\frac {\pi }{2}})}
f
′
(
x
)
=
−
sen
(
x
)
{\displaystyle f'(x)=-\operatorname {sen}(x)\,}
Derivada de la función tangente
editar
A partir de la regla del cociente , según la cual si la función que se quiere derivar,
f
(
x
)
{\displaystyle f(x)}
f
(
x
)
=
g
(
x
)
h
(
x
)
{\displaystyle f(x)={\frac {g(x)}{h(x)}}}
y
h
(
x
)
≠
0
{\displaystyle h(x)\neq 0\,}
g
(
x
)
/
h
(
x
)
{\displaystyle g(x)/h(x)\,}
d
d
x
f
(
x
)
=
f
′
(
x
)
=
g
′
(
x
)
h
(
x
)
−
g
(
x
)
h
′
(
x
)
[
h
(
x
)
]
2
{\displaystyle {\frac {d}{dx}}f(x)=f'(x)={\frac {g'(x)h(x)-g(x)h'(x)}{[h(x)]^{2}}}}
A partir de la identidad trigonométrica
tan
(
x
)
=
s
e
n
(
x
)
cos
(
x
)
+
{\displaystyle \tan(x)={sen(x) \over \cos(x)}+}
haciendo:
g
(
x
)
=
sen
(
x
)
{\displaystyle g(x)=\operatorname {sen}(x)\,}
g
′
(
x
)
=
cos
(
x
)
{\displaystyle g'(x)=\cos(x)\,}
h
(
x
)
=
cos
(
x
)
{\displaystyle h(x)=\cos(x)\,}
h
′
(
x
)
=
−
sen
(
x
)
{\displaystyle h'(x)=-\operatorname {sen}(x)\,}
sustituyendo resulta
f
′
(
x
)
=
cos
(
x
)
cos
(
x
)
−
sen
(
x
)
[
−
sen
(
x
)
]
cos
2
(
x
)
{\displaystyle f'(x)={\frac {\cos(x)\cos(x)-\operatorname {sen}(x)[-\operatorname {sen}(x)]}{\cos ^{2}(x)}}}
operando
f
′
(
x
)
=
cos
2
(
x
)
+
sen
2
(
x
)
cos
2
(
x
)
{\displaystyle f'(x)={\frac {\cos ^{2}(x)+\operatorname {sen} ^{2}(x)}{\cos ^{2}(x)}}}
y aplicando las identidades trigonométricas
cos
2
(
x
)
+
sen
2
(
x
)
=
1
{\displaystyle \cos ^{2}(x)+\operatorname {sen} ^{2}(x)=1\,}
sec
2
(
x
)
=
1
c
o
s
2
(
x
)
{\displaystyle \sec ^{2}(x)={\frac {1}{cos^{2}(x)}}\,}
resulta:
f
′
(
x
)
=
sec
2
(
x
)
{\displaystyle f'(x)=\sec ^{2}(x)\,}
Derivada de la función arcoseno
editar
Tenemos una función
y
=
a
r
c
s
e
n
x
{\displaystyle \,y=arcsen\,x}
sen
y
=
x
{\displaystyle \,\operatorname {sen} y=x}
cos
y
⋅
d
y
d
x
=
1
{\displaystyle \,\cos y\cdot {\frac {dy}{dx}}=1}
d
y
d
x
=
1
cos
y
{\displaystyle {\frac {dy}{dx}}={\frac {1}{\cos y}}\,}
Tenemos además que
cos
y
=
1
−
sen
2
y
{\displaystyle \cos y={\sqrt {1-\operatorname {sen} ^{2}y}}\,}
x
=
sen
y
{\displaystyle \,x=\operatorname {sen} y}
d
d
x
a
r
c
s
e
n
x
=
1
1
−
x
2
{\displaystyle \,{\frac {d}{dx}}arcsen\,x={\frac {1}{\sqrt {1-x^{2}}}}}
y
=
c
s
c
(
x
)
c
o
t
(
x
)
{\displaystyle y=csc(x)cot(x)\,}
y
′
=
(
−
c
s
c
(
x
)
c
s
c
2
(
x
)
)
−
c
o
t
(
x
)
c
s
c
(
x
)
c
o
t
(
x
)
{\displaystyle y'=(-csc(x)csc^{2}(x))-cot(x)csc(x)cot(x)\,}
y
′
=
−
c
s
c
(
x
)
c
s
c
2
(
x
)
−
c
o
t
2
(
x
)
c
s
c
(
x
)
{\displaystyle y'=-csc(x)csc^{2}(x)-cot^{2}(x)csc(x)\,}
y
′
=
−
c
s
c
3
(
x
)
−
c
o
t
2
(
x
)
c
s
c
(
x
)
{\displaystyle y'=-csc^{3}(x)-cot^{2}(x)csc(x)\,}
y
=
3
s
e
n
(
x
)
−
2
c
o
s
(
x
)
{\displaystyle y=3sen(x)-2cos(x)\,}
y
′
=
3
d
s
e
n
x
d
x
−
2
d
c
o
s
x
d
x
{\displaystyle y'=3{\frac {d\,senx}{dx}}-2{\frac {d\,cosx}{dx}}}
y
′
=
3
c
o
s
(
x
)
+
2
s
e
n
(
x
)
{\displaystyle y'=3cos(x)+2sen(x)\,}
![{\displaystyle {\frac {d}{dx}}f(x)=f'(x)={\frac {g'(x)h(x)-g(x)h'(x)}{[h(x)]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c88a63b6dca6feb28c3fbfa5db1925eb400bfd4)
![{\displaystyle f'(x)={\frac {\cos(x)\cos(x)-\operatorname {sen}(x)[-\operatorname {sen}(x)]}{\cos ^{2}(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a024f3b34dcb44927916a83bda03c94850dd14e9)
Datos: Q3296857