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on: Derivative Techniques 2 j - Derivatives of Trig Functio

Lesson: Derivative Techniques 2 Obj - Derivatives of Trig Functions

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Page 1: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Lesson: Derivative Techniques 2

Obj - Derivatives of Trig Functions

Page 2: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Trig Essentials SOHCAHTOA

siny r

cscr y

cosx r

secr x

tan coty x

x y

y

x

opp(y)

adj(x)

hyp(r)

Page 3: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Now, let’s examine the trig function graphs

sin( )y x

cos( )y x

tan( )y x

Page 4: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

sin( )y x

Page 5: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

cos( )y x

Page 6: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

tan( )y x

Page 7: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions
Page 8: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Practice!

Page 9: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Other trig values

0

(1,0)

rads

(0,1)2

rads

( 1,0)rads

3(0, 1)

2rads

Page 10: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions
Page 11: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Basic Trig Identities1

csc xsinx

1sec

cosx

x

1cot

tanx

x 2 2sin cos 1x x

coscot

sin

xx

x

sintan

cos

xx

x

2 2cos 1 sinx x 2 2sin 1 cosx x

sin(2 ) 2 sin cosx x x

Page 12: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Trig Derivative Equations:

1.

2.

Page 13: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

What about Tangent, Cotangent, Secant, and Cosecant?

Let’s derive them!

dTanx

dx

d Sinx

dx Cosx

Use quotient rule:'

2

' 'f g f f g

g g

Page 14: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

2

( ) ( )

( )

Cosx Cosx Sinx Sinx

Cosx

2 2

2

Cos x Sin x

Cos x

2

1

Cos x

Page 15: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

21

Cosx

2Sec x

Therefore,

Page 16: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

Now try:

dCotx

dx

dSecx

dx

dCscx

dx

Page 17: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

So,

1. 2.

3. 4.

5. 6.

Page 18: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

O.K., How do I remember all that?

Look at the cofunctions

The derivative of any cofunctioncan be obtained from the derivativeof the corresponding function bytaking the negation and replacingeach function in the derivative withits cofunction.

Page 19: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

EX. 1: Find dy/dx if y x Sinx

Page 20: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

EX. 2: Find dy/dx if1

Sinxy

Cosx

Page 21: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

EX.3 : Find '' ( )4

f if f x Secx

Page 22: Lesson: Derivative Techniques 2  Obj - Derivatives of Trig Functions

EX. 4: Find

87

87( ) [ ]

da Sinx

dx

100

100( ) [ ]

db Cosx

dx