Derivatives of Trig Functions

Objective: Memorize the derivatives of the six trig functions

Derivative of the sin(x)

• The derivative of the sinx is:

xxdx

dcos][sin

Derivative of the sin(x)

• The derivative of the sinx is:• Lets look at the two graphs together.

xxdx

dcos][sin

Derivative of the cos(x)

• The derivative of the cosx is:

xxdx

dsin][cos

Derivative of the cos(x)

• The derivative of the cosx is:• Lets look at the two graphs together.

xxdx

dsin][cos

Derivatives of trig functions

• The derivatives of all six trig functions:

xxdx

d 2sec][tan

xxdx

dsin][cos xx

dx

dcos][sin

xxdx

d 2csc][cot

xxdx

dtansec[sec] xxx

dx

dcotcsc][csc

Trig Identities

1cossin 22 xx

xx 22 sectan1

xx 22 csccot1

cossin22sin

22 sincos2cos

Example 1

• Find if dx

dyxxy sin

Example 1

• Find if

• We need to use the product rule to solve.

dx

dyxxy sin

)1(sincos xxxdx

dy

Example 2

• Find if dx

dy

x

xy

cos1

sin

Example 2

• Find if

• We need to use the quotient rule to solve.

dx

dy

x

xy

cos1

sin

2)cos1(

)sin)((sin))(coscos1(

x

xxxx

dx

dy

Example 2

• Find if

• We need to use the quotient rule to solve.

dx

dy

x

xy

cos1

sin

2)cos1(

)sin)((sin))(coscos1(

x

xxxx

dx

dy

xx

x

x

xxx

dx

dy

cos1

1

)cos1(

1cos

)cos1(

sincoscos22

22

Example 3

• Find if . )4/(// f xxf sec)(

Example 3

• Find if . )4/(// f xxf sec)(

xxxf tansec)(/

Example 3

• Find if . )4/(// f xxf sec)(

xxxf tansec)(/

xxxxxxf tansectansecsec)( 2//

xxxxf sectansec)( 23//

Example 3

• Find if . )4/(// f xxf sec)(

xxxf tansec)(/

xxxxxxf tansectansecsec)( 2//

xxxxf sectansec)( 23//

)4/sec()4/(tan)4/(sec)4/( 23// f

23212)4/( 23// f

Example 4

• On a sunny day, a 50-ft flagpole casts a shadow that changes with the angle of elevation of the Sun. Let s be the length of the shadow and the angle of elevation of the Sun. Find the rate at which the shadow is changing with respect to when .

045

Example 4

• On a sunny day, a 50-ft flagpole casts a shadow that changes with the angle of elevation of the Sun. Let s be the length of the shadow and the angle of elevation of the Sun. Find the rate at which the shadow is changing with respect to when .

• The variables s and are related by or .

045 s/50tan

cot50s

Example 4

• We are looking for the rate of change of s with respect to . In other words, we are looking to solve for . In this example, is the independent var.

dds

Example 4

• We are looking for the rate of change of s with respect to . In other words, we are looking to solve for . In this example, is the independent var.

dds

cot50s

2csc50d

ds)4/(csc50 2

d

ds

Example 4

• We are looking for the rate of change of s with respect to . In other words, we are looking to solve for . In this example, is the independent var.

dds

cot50s

2csc50d

ds)4/(csc50 2

d

ds

radianftd

ds/100

deg/75.1

deg9

5

deg180100 ft

ftrad

rad

ft

Class work

• Section 2.5• Page 172• 2-16 even

Homework

• Section 2.5• Page 172• 1-27 odd• 31