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Función
Derivada
sen
(
x
)
{\displaystyle \operatorname {sen}(x)}
cos
(
x
)
{\displaystyle \cos(x)}
cos
(
x
)
{\displaystyle \cos(x)}
−
sen
(
x
)
{\displaystyle -\operatorname {sen}(x)}
tan
(
x
)
{\displaystyle \tan(x)}
sec
2
(
x
)
{\displaystyle \sec ^{2}(x)}
cot
(
x
)
{\displaystyle \cot(x)}
−
csc
2
(
x
)
{\displaystyle -\csc ^{2}(x)}
sec
(
x
)
{\displaystyle \sec(x)}
sec
(
x
)
tan
(
x
)
{\displaystyle \sec(x)\tan(x)}
csc
(
x
)
{\displaystyle \csc(x)}
−
csc
(
x
)
cot
(
x
)
{\displaystyle -\csc(x)\cot(x)}
arcsen
(
x
)
{\displaystyle \operatorname {arcsen} (x)}
1
1
−
x
2
{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}
arccos
(
x
)
{\displaystyle \arccos(x)}
−
1
1
−
x
2
{\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}
arctan
(
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)}
[ editar ] Dada la función
f
(
x
)
=
cos
(
x
)
=
sen
(
x
+
π
2
)
{\displaystyle f(x)=\cos(x)=\operatorname {sen} \left(x+{\frac {\pi }{2}}\right)}
f
′
(
x
)
=
−
sen
(
x
)
{\displaystyle f'(x)=-\operatorname {sen}(x)}
[ 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
)
=
sen
(
x
)
cos
(
x
)
{\displaystyle \tan(x)={\frac {\operatorname {sen}(x)}{\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
cos
2
(
x
)
{\displaystyle \sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}}
resulta:
f
′
(
x
)
=
sec
2
(
x
)
{\displaystyle f'(x)=\sec ^{2}(x)}
[ editar ] Tenemos una función
y
=
arcsen
x
{\displaystyle y=\operatorname {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
arcsen
x
=
1
1
−
x
2
{\displaystyle {\frac {d}{dx}}\operatorname {arcsen} x={\frac {1}{\sqrt {1-x^{2}}}}}
y
=
csc
(
x
)
cot
(
x
)
{\displaystyle y=\csc(x)\cot(x)}
y
′
=
−
csc
(
x
)
csc
2
(
x
)
−
cot
(
x
)
csc
(
x
)
cot
(
x
)
{\displaystyle y'=-\csc(x)\csc ^{2}(x)-\cot(x)\csc(x)\cot(x)}
y
′
=
−
csc
(
x
)
csc
2
(
x
)
−
cot
2
(
x
)
csc
(
x
)
{\displaystyle y'=-\csc(x)\csc ^{2}(x)-\cot ^{2}(x)\csc(x)}
y
′
=
−
csc
3
(
x
)
−
cot
2
(
x
)
csc
(
x
)
{\displaystyle y'=-\csc ^{3}(x)-\cot ^{2}(x)\csc(x)}
y
=
3
sen
(
x
)
−
2
cos
(
x
)
{\displaystyle y=3\operatorname {sen}(x)-2\cos(x)}
y
′
=
3
d
sen
x
d
x
−
2
d
cos
x
d
x
{\displaystyle y'=3{\frac {d\operatorname {sen} x}{dx}}-2{\frac {d\cos x}{dx}}}
y
′
=
3
cos
(
x
)
+
2
sen
(
x
)
{\displaystyle y'=3\cos(x)+2\operatorname {sen}(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