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