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)

Now, let’s examine the trig function graphs

sin( )y x

cos( )y x

tan( )y x

sin( )y x

cos( )y x

tan( )y x

Practice!

Other trig values

0

(1,0)

rads

(0,1)2

rads

( 1,0)rads

3(0, 1)

2rads

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

Trig Derivative Equations:

1.

2.

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

2

( ) ( )

( )

Cosx Cosx Sinx Sinx

Cosx

2 2

2

Cos x Sin x

Cos x

2

1

Cos x

21

Cosx

2Sec x

Therefore,

Now try:

dCotx

dx

dSecx

dx

dCscx

dx

So,

1. 2.

3. 4.

5. 6.

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.

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

EX. 2: Find dy/dx if1

Sinxy

Cosx

EX.3 : Find '' ( )4

f if f x Secx

EX. 4: Find

87

87( ) [ ]

da Sinx

dx

100

100( ) [ ]

db Cosx

dx