Download Derivatives of Trig Functions

Derivatives of Trig Functions
Objective: Memorize the derivatives
of the six trig functions
Derivative of the sin(x)
• The derivative of the sinx is:
d
[sin x]  cos x
dx
Derivative of the sin(x)
• The derivative of the sinx is:
• Lets look at the two graphs together.
d
[sin x]  cos x
dx
Derivative of the cos(x)
• The derivative of the cosx is:
d
[cos x]   sin x
dx
Derivative of the cos(x)
• The derivative of the cosx is:
• Lets look at the two graphs together.
d
[cos x]   sin x
dx
Derivatives of trig functions
• The derivatives of all six trig functions:
d
[sin x]  cos x
dx
d
[cos x]   sin x
dx
d
[tan x]  sec 2 x
dx
d
[cot x]   csc 2 x
dx
d
[sec]  sec x tan x
dx
d
[csc x]   csc x cot x
dx
Trig Identities
sin 2 x  cos 2 x  1
1  tan x  sec x
2
2
1  cot 2 x  csc 2 x
sin 2  2 sin  cos
Example 1
• Find
dy
dx
if y  x sin x
Example 1
• Find
dy
dx
if y  x sin x
• We need to use the product rule to solve.
dy
 x cos x  sin x(1)
dx
Example 2
• Find
dy
dx
if
sin x
y
1  cos x
Example 2
• Find
dy
dx
if
sin x
y
1  cos x
• We need to use the quotient rule to solve.
dy (1  cos x)(cos x)  (sin x)(  sin x)

dx
(1  cos x) 2
Example 2
• Find
dy
dx
if
sin x
y
1  cos x
• We need to use the quotient rule to solve.
dy (1  cos x)(cos x)  (sin x)(  sin x)

dx
(1  cos x) 2
dy cos x  cos 2 x  sin 2 x
cos x  1
1



2
2
dx
(1  cos x)
(1  cos x) 1  cos x
Example 3
//
• Find f ( / 4) if
f ( x)  sec x .
Example 3
//
• Find f ( / 4) if
f ( x)  sec x .
f ( x)  sec x tan x
/
Example 3
//
• Find f ( / 4) if
f ( x)  sec x .
f ( x)  sec x tan x
/
f // ( x)  sec x sec 2 x  tan x sec x tan x
f // ( x)  sec3 x  tan 2 x sec x
Example 3
//
• Find f ( / 4) if
f ( x)  sec x .
f ( x)  sec x tan x
/
f // ( x)  sec x sec 2 x  tan x sec x tan x
f // ( x)  sec3 x  tan 2 x sec x
f // ( / 4)  sec 3 ( / 4)  tan 2 ( / 4) sec( / 4)
f ( / 4) 
//
 2   1  2   3
3
2
2
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   450.
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   450.
• The variables s and  are related by tan   50 / s
or s  50 cot  .
Example 4
• We are looking for the rate of change of s with
respect to  . In other words, we are looking to solve
for ds . In this example,  is the independent var.
d
Example 4
• We are looking for the rate of change of s with
respect to  . In other words, we are looking to solve
for ds . In this example,  is the independent var.
d
s  50 cot 
ds
 50 csc 2 
d
ds
 50 csc 2 ( / 4)
d
Example 4
• We are looking for the rate of change of s with
respect to  . In other words, we are looking to solve
for ds . In this example,  is the independent var.
d
s  50 cot 
ds
 50 csc 2 
d
ds
 100 ft / radian
d
ds
 50 csc 2 ( / 4)
d
 100
ft  rad
5
ft

 
 1.75 ft / deg
rad 180 deg
9 deg
Class work
• Section 2.5
• Page 172
• 2-16 even
Homework
• Section 2.5
• Page 172
• 1-25 odd