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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