4.2.69

Demuestre por medio de la definición

d

d
x

sen
⁡
# x
cos
⁡
x

{\displaystyle {\frac {d}{dx}}\operatorname {sen} x=\cos x}

Ocupando la definición:

d

d
x

f
(
x
# )

lim

Δ
x
→
0

f
(
x
Δ
x
)
−
f
(
x
)

Δ
x

{\displaystyle \frac{d}{dx} f(x) = \lim\_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}}

d

d
x

sen
⁡
# x

lim

Δ
x
→
0

sen
⁡
(
x
Δ
x
)
−
sen
⁡
(
x
)

Δ
x

{\displaystyle {\frac {d}{dx}}\operatorname {sen} x=\lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(x+\Delta x)-\operatorname {sen}(x)}{\Delta x}}}

Aplicando la siguiente igualdad

sen
⁡
(
a
b
# )
sen
⁡
a
cos
⁡
b
sen
⁡
b
cos
⁡
a

{\displaystyle \operatorname {sen}(a+b)=\operatorname {sen} a\cos b+\operatorname {sen} b\cos a}

lim

Δ
x
→
0

sen
⁡
(
x
Δ
x
)
−
sen
⁡
(
x
)

Δ
x

=

lim

Δ
x
→
0

(
(
sen
⁡
(
x
)
cos
⁡
(
Δ
x
)
)
(
sen
⁡
(
Δ
x
)
cos
⁡
(
x
)
)
−
sen
⁡
(
x
)

Δ
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(x+\Delta x)-\operatorname {sen}(x)}{\Delta x}}=\lim \_{\Delta x\rightarrow 0}{\frac {((\operatorname {sen}(x)\cos(\Delta x))+(\operatorname {sen}(\Delta x)\cos(x))-\operatorname {sen}(x)}{\Delta x}}}

Acomodando con términos semejantes;

lim

Δ
x
→
0

(
(
sen
⁡
(
x
)
cos
⁡
(
Δ
x
)
)
−
sen
⁡
(
x
)

Δ
x

(
sen
⁡
(
Δ
x
)
cos
⁡
(
x
)
)

Δ
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}{\frac {((\operatorname {sen}(x)\cos(\Delta x))-\operatorname {sen}(x)}{\Delta x}}+{\frac {(\operatorname {sen}(\Delta x)\cos(x))}{\Delta x}}}

lim

Δ
x
→
0

(
(
sen
⁡
(
x
)
cos
⁡
(
Δ
x
)
)
−
sen
⁡
(
x
)

Δ
x

lim

Δ
x
→
0

(
sen
⁡
(
Δ
x
)
cos
⁡
(
x
)
)

Δ
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}{\frac {((\operatorname {sen}(x)\cos(\Delta x))-\operatorname {sen}(x)}{\Delta x}}+\lim \_{\Delta x\rightarrow 0}{\frac {(\operatorname {sen}(\Delta x)\cos(x))}{\Delta x}}}

Factorizando

sen
⁡
x

{\displaystyle \operatorname {sen} x}

y

cos
⁡
x

{\displaystyle \cos x}

:

lim

Δ
x
→
0

sen
⁡
x

cos
⁡
(
Δ
x
)
−
1

Δ
x

lim

Δ
x
→
0

cos
⁡
x

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}\operatorname {sen} x{\frac {\cos(\Delta x)-1}{\Delta x}}+\lim \_{\Delta x\rightarrow 0}\cos x{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

Ocupando la propiedad del producto:

[

lim

Δ
x
→
0

sen
⁡
x

lim

Δ
x
→
0

cos
⁡
(
Δ
x
)
−
1

Δ
x

]
[

lim

Δ
x
→
0

cos
⁡
x

lim

Δ
x
→
0

sen
⁡
(
Δ
x
)

Δ
x

]

{\displaystyle [\lim \_{\Delta x\rightarrow 0}\operatorname {sen} x\lim \_{\Delta x\rightarrow 0}{\frac {\cos(\Delta x)-1}{\Delta x}}]+[\lim \_{\Delta x\rightarrow 0}\cos x\lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}]}

lim

Δ
x
→
0

sen
⁡
# x
sen
⁡
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}\operatorname {sen} x=\operatorname {sen} x}

lim

Δ
x
→
0

cos
⁡
# x
cos
⁡
x

{\displaystyle \lim\_{\Delta x \rightarrow 0} \cos x = \cos x}

* sen
⁡
(
x
)

lim

Δ
x
→
0

cos
⁡
(
Δ
x
)
−
1

Δ
x

cos
⁡
(
x
)

lim

Δ
x
→
0

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle \operatorname {sen}(x)\lim \_{\Delta x\rightarrow 0}{\frac {\cos(\Delta x)-1}{\Delta x}}+\cos(x)\lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

Evaluando el limite de las siguientes funciones

lim

Δ
x
→
0

cos
⁡
(
Δ
x
)
−
1

Δ
x

{\displaystyle \lim\_{\Delta x \rightarrow 0} \frac{\cos(\Delta x) - 1} {\Delta x}}

;

lim

Δ
x
→
0

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle \lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

:

sen
⁡
(
x
)

lim

Δ
x
→
0

cos
⁡
(
Δ
x
)
−
1

Δ
x

cos
⁡
(
x
)

lim

Δ
x
→
0

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle \operatorname {sen}(x)\lim \_{\Delta x\rightarrow 0}{\frac {\cos(\Delta x)-1}{\Delta x}}+\cos(x)\lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

Grafica de la función de

f
(
x
# )

cos
⁡
(
Δ
x
)
−
1

Δ
x

{\displaystyle f(x) = \frac{\cos(\Delta x) - 1} {\Delta x}}

Grafica de la función de

f
(
x
# )

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle f(x)={\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

sen
⁡
(
x
)

lim

Δ
x
→
0

cos
⁡
(
Δ
x
)
−
1

Δ
x

cos
⁡
(
x
)

lim

Δ
x
→
0

sen
⁡
(
Δ
x
)

Δ
x

{\displaystyle \operatorname {sen}(x)\lim \_{\Delta x\rightarrow 0}{\frac {\cos(\Delta x)-1}{\Delta x}}+\cos(x)\lim \_{\Delta x\rightarrow 0}{\frac {\operatorname {sen}(\Delta x)}{\Delta x}}}

sen
⁡
(
x
)
(
0
)
cos
⁡
(
x
)
(
1
)

{\displaystyle \operatorname {sen}(x)(0)+\cos(x)(1)}

d

d
x

sen
⁡
# x
cos
⁡
x

{\displaystyle {\frac {d}{dx}}\operatorname {sen} x=\cos x}