Drill

• Convert 105 degrees to radians

• Convert 5π/9 to radians

• What is the range of the equation y = 2 + 4cos3x?

• 7π/12

• 100 degrees

• [-2, 6]

Derivatives of Trigonometric Functions

Lesson 3.5

Objectives

• Students will be able to– use the rules for differentiating the six basic

trigonometric functions.

Find the derivative of the sine function.

xy sin h

xfhxfxf

h

0lim'

H

xHxy

H

sinsinlim'

0

H

xHxHxy

H

sinsincoscossinlim'

0

H

HxxHxy

H

sincossincossinlim'

0

H

HxHxy

H

sincos1cossinlim'

0

Find the derivative of the sine function.

xy sin h

xfhxfxf

h

0lim'

H

xHxy

H

sinsinlim'

0

H

xHxHxy

H

sinsincoscossinlim'

0

H

HxxHxy

H

sincossincossinlim'

0

H

HxHxy

H

sincos1cossinlim'

0

H

Hx

H

Hxy

HH

sincoslim

1cossinlim'

00

H

Hx

H

Hxy

HH

sinlimcos

1coslimsin'

00

1cos0sin' xxy

xy cos'

Find the derivative of the cosine function.xy cos

h

xfhxfxf

h

0lim'

H

xHxy

H

coscoslim'

0

H

xHxHxy

H

cossinsincoscoslim'

0

H

HxxHxy

H

sinsincoscoscoslim'

0

H

HxHxy

H

sinsin1coscoslim'

0

Find the derivative of the cosine function.xy cos

H

HxHxy

H

sinsin1coscoslim'

0

H

Hx

H

Hxy

HH

sinsinlim

1coscoslim'

00

H

Hx

H

Hxy

HH

sinlimsin

1coslimcos'

00

1sin0cos' xxy

xy sin'

Derivatives of Trigonometric Functions

xxdx

dcossin

xxdx

dsincos

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy cos3

33 coscos xdx

dxx

dx

dx

dx

dy

23 3cossin xxxxdx

dy

xxxxdx

dysincos3 32

Example 1 Differentiating with Sine and Cosine

Find the derivative.

x

xy

cos2

sin

2cos2

cos2sinsincos2

x

xdx

dxx

dx

dx

dx

dy

2cos2

sin0sincoscos2

x

xxxx

dx

dy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

x

xy

cos2

sin

2cos2

sin0sincoscos2

x

xxxx

dx

dy

2cos2

sinsincoscos2

x

xxxx

dx

dy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

x

xy

cos2

sin

2cos2

sinsincoscos2

x

xxxx

dx

dy

2

22

cos2

sincoscos2

x

xxx

dx

dy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

x

xy

cos2

sin

2

22

cos2

sincoscos2

x

xxx

dx

dy

2

22

cos2

cos1coscos2

x

xxx

dx

dy

Remember that cos2 x + sin2 x = 1So sin x = 1 – cos 2x

Example 1 Differentiating with Sine and Cosine

Find the derivative.

x

xy

cos2

sin

2

22

cos2

cos1coscos2

x

xxx

dx

dy

2cos2

1cos2

x

x

dx

dy

Homework, day #1

• Page 146: 1-3, 5, 7, 8, 10• On 13 – 16

Velocity is the 1st derivative Speed is the absolute value of velocity Acceleration is the 2nd derivative Look at the original function to determine

motion

Find the derivative of the tangent function.xy tan

x

xy

cos

sin

x

xdx

dxx

dx

dx

y2cos

cossinsincos'

x

xxxxy

2cos

sinsincoscos'

x

xxy

2

22

cos

sincos'

Find the derivative of the tangent function.xy tan

x

xxy

2

22

cos

sincos'

xy

2cos

1'

2

cos

1'

xy

xy 2sec'

Derivatives of Trigonometric Functions

xxdx

dcossin

xxdx

dsincos

xxdx

d 2sectan

Derivatives of Trigonometric Functions

xxdx

dcossin

xxdx

dsincos

xxdx

d 2sectan xxdx

d 2csccot

xxxdx

dtansecsec

xxxdx

dcotcsccsc

More Examples with Trigonometric FunctionsFind the derivative of y.

xxy cot11sin

1sincot1cot11sin xdx

dxx

dx

dx

dx

dy

xxxxdx

dycoscot1csc1sin 2

xx

x

xx

dx

dycos

sin

cos1

sin

11sin

2

x

xx

xxdx

dy

sin

coscos

sin

1

sin

1 2

2

xxxxxdx

dycsccoscoscsccsc 22

xxxxxdx

dycsccoscoscsccsc 22

xxxxxdx

dy 22 csccoscsccoscsc

xxxxdx

dy 22 csccos)cos1(csc

xxxxdx

dy 22 csccos)(sincsc

xxxxdx

dy 22 csccos)(sinsin

1

xxxdx

dy 2csccossin

More Examples with Trigonometric Functions

Find the derivative of y.

3

3

sec

2tan

xx

xxy

23

3333

sec

sec2tan2tansec

xx

xxdx

dxxxx

dx

dxx

dx

dy

23

23223

sec

3tansec2tan6secsec

xx

xxxxxxxxx

dx

dy

52323

223

6secsec6sec

6secsec

xxxxxx

xxxx

5322

23

6tansec2tan3tansec

3tansec2tan

xxxxxxxx

xxxxx

)6tansec2tan3tan(sec6secsec6sec 532252323 xxxxxxxxxxxxxx

xxxxxxxxxxxx tansec2tan3tansecsecsec6sec 3222323

23

3222323

sec

tansec2tan3tansecsecsec6sec'

xx

xxxxxxxxxxxxy

Whatta Jerk!

Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is

.3

3

dt

sd

dt

datj

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos t

s2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

tts cos31

tdt

dssin31 tsin3

tdt

sdcos3

21

2

velocity

acceleration

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos t

s2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

tts cos31

tdt

dssin31 tsin3

tdt

sdcos3

21

2

velocity

acceleration

tdt

sdsin3

31

3

tsin3jerk

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos t

s2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

ttts cossin22

ttttdt

dssincos2sincos21

ttttdt

sdcossin2cossin2

21

2

velocity

acceleration

ttttdt

sdsincos2sincos2

31

3

jerk

Homework, day #2

• Page 146: 4, 6, 9, 11, 12, 17-20, 22 28, 32