Derivatives of Trigonometric Functions, Part 1
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Transcript of Derivatives of Trigonometric Functions, Part 1
![Page 1: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/1.jpg)
![Page 2: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/2.jpg)
![Page 3: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/3.jpg)
Derivatives of Trigonometric Functions
1. f (x) = sin x
2. g(x) = cos x
![Page 4: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/4.jpg)
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
![Page 5: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/5.jpg)
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
![Page 6: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/6.jpg)
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
![Page 7: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/7.jpg)
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 .
![Page 8: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/8.jpg)
The derivative of sin x
![Page 9: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/9.jpg)
The derivative of sin x
First of all, there is a little trig identity we need to remember:
![Page 10: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/10.jpg)
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
![Page 11: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/11.jpg)
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.
![Page 12: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/12.jpg)
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:
![Page 13: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/13.jpg)
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
![Page 14: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/14.jpg)
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
![Page 15: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/15.jpg)
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):
![Page 16: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/16.jpg)
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
![Page 17: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/17.jpg)
The derivative of sin x
![Page 18: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/18.jpg)
The derivative of sin x
So, we need to solve this limit:
![Page 19: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/19.jpg)
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
![Page 20: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/20.jpg)
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:
![Page 21: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/21.jpg)
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
![Page 22: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/22.jpg)
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
![Page 23: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/23.jpg)
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
![Page 24: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/24.jpg)
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
![Page 25: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/25.jpg)
The derivative of sin x
![Page 26: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/26.jpg)
The derivative of sin x
So, we have that:
![Page 27: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/27.jpg)
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
![Page 28: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/28.jpg)
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=
![Page 29: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/29.jpg)
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=
![Page 30: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/30.jpg)
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=
![Page 31: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/31.jpg)
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)
![Page 32: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/32.jpg)
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)
![Page 33: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/33.jpg)
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
![Page 34: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/34.jpg)
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
![Page 35: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/35.jpg)
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
![Page 36: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/36.jpg)
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
![Page 37: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/37.jpg)
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
![Page 38: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/38.jpg)
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
![Page 39: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/39.jpg)
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
![Page 40: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/40.jpg)
The derivative of sin x
![Page 41: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/41.jpg)
The derivative of sin x
So, finally:
![Page 42: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/42.jpg)
The derivative of sin x
So, finally:
f ′(x) = cos x
![Page 43: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/43.jpg)
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
![Page 44: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/44.jpg)
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
d
dx(sin x) = cos x
![Page 45: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/45.jpg)
The derivative of sin x
So, finally:
f ′(x) = cos x
Another way to put it:
d
dx(sin x) = cos x
![Page 46: Derivatives of Trigonometric Functions, Part 1](https://reader034.fdocuments.in/reader034/viewer/2022050905/5482f31cb4af9f910d8b48af/html5/thumbnails/46.jpg)