Post on 05-Dec-2014
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Derivatives of Trigonometric Functions
1. f (x) = sin x
2. g(x) = cos x
Derivatives of Trigonometric Functions
The derivative of any trigonometric function can be found once weknow the derivative of the two basic functions:
1. f (x) = sin x
2. g(x) = cos x
Derivatives of Trigonometric Functions
The derivative of any trigonometric function can be found once weknow the derivative of the two basic functions:
1. f (x) = sin x
2. g(x) = cos x
Derivatives of Trigonometric Functions
The derivative of any trigonometric function can be found once weknow the derivative of the two basic functions:
1. f (x) = sin x
2. g(x) = cos x
Derivatives of Trigonometric Functions
The derivative of any trigonometric function can be found once weknow the derivative of the two basic functions:
1. f (x) = sin x
2. g(x) = cos x
In this video we’re going to find the derivative of sin x .
The derivative of sin x
The derivative of sin x
First of all, there is a little trig identity we need to remember:
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.By definition:
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.By definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.By definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x= lim
∆x→0
sin(x + ∆x) − sin x
∆x
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.By definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x= lim
∆x→0
sin(x + ∆x) − sin x
∆x
Now we use the trig identity to expand sin(x + ∆x):
The derivative of sin x
First of all, there is a little trig identity we need to remember:
sin(a + b) = sin a cos b + sin b cos a
Let’s now consider the function f (x) = sin x and let’s find itsderivative.By definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x= lim
∆x→0
sin(x + ∆x) − sin x
∆x
Now we use the trig identity to expand sin(x + ∆x):
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
The derivative of sin x
The derivative of sin x
So, we need to solve this limit:
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
We can factor in the numerator:
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
We can factor in the numerator:
f ′(x) = lim∆x→0
sin x (cos ∆x − 1) + sin ∆x cos x
∆x
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
We can factor in the numerator:
f ′(x) = lim∆x→0
sin x (cos ∆x − 1) + sin ∆x cos x
∆x
= sin x lim∆x→0
cos ∆x − 1
∆x+ cos x lim
∆x→0
sin ∆x
∆x
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
We can factor in the numerator:
f ′(x) = lim∆x→0
sin x (cos ∆x − 1) + sin ∆x cos x
∆x
= sin x lim∆x→0
cos ∆x − 1
∆x+ cos x
�������*
1
lim∆x→0
sin ∆x
∆x
The derivative of sin x
So, we need to solve this limit:
f ′(x) = lim∆x→0
sin x cos ∆x + sin ∆x cos x − sin x
∆x
We can factor in the numerator:
f ′(x) = lim∆x→0
sin x (cos ∆x − 1) + sin ∆x cos x
∆x
= sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x��
�����*
1
lim∆x→0
sin ∆x
∆x
The derivative of sin x
The derivative of sin x
So, we have that:
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x=
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x=
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
1 − cos2 ∆x
∆x (1 + cos ∆x)
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= − lim∆x→0
sin ∆x
∆x. lim
∆x→0
sin ∆x
1 + cos ∆x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −���
����*1
lim∆x→0
sin ∆x
∆x. lim
∆x→0
sin ∆x
1 + cos ∆x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −���
����*1
lim∆x→0
sin ∆x
∆x. lim
∆x→0
����: 0
sin ∆x
1 + cos ∆x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −���
����*1
lim∆x→0
sin ∆x
∆x. lim
∆x→0
����: 0
sin ∆x
1 +����: 1
cos ∆x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −���
����*1
lim∆x→0
sin ∆x
∆x.���
������:
0lim
∆x→0
sin ∆x
1 + cos ∆x
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −���
����*1
lim∆x→0
sin ∆x
∆x.���
������:
0lim
∆x→0
sin ∆x
1 + cos ∆x= 0
The derivative of sin x
So, we have that:
f ′(x) = sin x lim∆x→0
cos ∆x − 1
∆x︸ ︷︷ ︸We’ll show this is 0!
+ cos x
lim∆x→0
cos ∆x − 1
∆x= − lim
∆x→0
1 − cos ∆x
∆x=
= − lim∆x→0
1 − cos ∆x
∆x.1 + cos ∆x
1 + cos ∆x= − lim
∆x→0
�����
��: sin2 ∆x
1 − cos2 ∆x
∆x (1 + cos ∆x)
= −�������*1
lim∆x→0
sin ∆x
∆x.���
������:
0lim
∆x→0
sin ∆x
1 + cos ∆x= 0
The derivative of sin x
The derivative of sin x
So, finally:
The derivative of sin x
So, finally:
f ′(x) = cos x
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
d
dx(sin x) = cos x
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
d
dx(sin x) = cos x