Upload
bertram-wade
View
231
Download
7
Tags:
Embed Size (px)
Citation preview
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