5
Logarithmic, Exponential, and
Other Transcendental Functions
If you aren't in over your
head, how do you know
how tall you are?
T. S. Eliot
Copyright © Cengage Learning. All rights reserved.
5.6
Inverse Trigonometric
Functions: Differentiation
Copyright © Cengage Learning. All rights reserved.
Inverse Trigonometric Functions
Under suitable restrictions, each of the six trigonometric
functions is one-to-one and so has an inverse function, as
shown in the following definition.
9
Inverse Trigonometric Functions
The graphs of the six inverse trigonometric functions are
shown in Figure 5.29.
Figure 5.29
10
Example 1 – Evaluating Inverse Trigonometric Functions
Evaluate each function.
Solution:
11
Example 1 – Evaluating Inverse Trigonometric Functions
cont’d
12
Inverse Trigonometric Functions
Inverse functions have the properties
f(f –1(x)) = x and f –1(f(x)) = x.
When applying these properties to inverse trigonometric
functions, remember that the trigonometric functions have
inverse functions only in restricted domains.
For x-values outside these domains, these two properties
do not hold.
For example, arcsin(sin π) is equal to 0, not π.
13
Inverse Trigonometric Functions
14
Example 2 – Solving an Equation
15
Inverse Trig Functions
Given y  arcsin x, where 0  y   / 2, find cos y.
sin y  x
1
x
y
1 x
cos y  1  x
2
2
16
Inverse Trig Functions
Given y  arcsec

5
sec y 
2

5 / 2 , find tan y.
5
y
1
2
tan y  1/ 2
17
Inverse Trig Functions
 1 x 
Find sec  tan

3

x
tan y 
3
x
y
3
2
x
9

x


sec  tan 1  
3
3

18
Derivatives of Inverse Trigonometric
Functions
19
Inverse Trig Functions and Differentiation
Find y
1
y  sin u,
sin y  u
1
u
Take derivative implicitly
y
 cos y  y  u
u
y 
 cos y 
?
y 
u
2
x
9

x
2 tan 1  
sec
1  u 

3
3
20
Inverse Trig Functions and Differentiation
1
y  tan u,
tan y  u
Find y
?
u
Take derivative implicitly
y
2
sec
 y  y  u 
u
y 

2
sec y
1

u
u2 1

2
2
x
9

x



sec  tan y1   u
3 2 3

u 1
21
Derivatives of Inverse Trigonometric Functions
The following theorem lists the derivatives of the six inverse
trigonometric functions.
22
Inverse Trig Functions and Differentiation
2
d
arcsin 2 x  
2
dx
1  4x
d
u'
arcsin u  
2
dx
1 u
d
3
 arctan 3x   1  9x 2
dx
d
u'
arctan u  
dx
1  u2
24
Derivatives of Inverse Trigonometric Functions
d 

arcsin
x

dx 
d
u'
arcsin u  
2
dx
1 u
1 / 2x 1/ 2
1
1


1 x
2 x 1  x 2 x  x2
25
Derivatives of Inverse Trigonometric Functions
d
2x
 arc sec e  
dx
2e
e
2x
d
u'
arcsec u  
2
dx
u u 1
2x
(e )  1
2x 2

2
e 1
4x
26
Derivatives of Inverse Trigonometric Functions
Differentiate y  arcsin x  x 1  x 2
d
u'
arcsin u  
dx
1  u2
1
2 1/ 2
y'
 1  x  x   1  x   2 x 
2
2
1 x
1

11 x
1 x
2
2
2

x
2
1 x
2

2 1  x
2
1 x
2
2
1 x
2
27
Review of Basic Differentiation
Rules
29
Summary of Differentiation Rules
1.
2.
3.
4.
5.
6.
7.
8.
d
cu   cu '
dx
d
u  v   u ' v '
dx
d
uv   u ' v  uv '
dx
d  u  u ' v  uv '



dx  v 
v2
d
c  0
dx
d n
u   nu n 1u '
dx
d
 x  1
dx
d
u
 u    u ' , u  0
dx
u
d
u'
ln
u

 
dx
u
d u
e   eu u '
10.
dx
d
u'
11.
log
u

 a 
dx
 ln a  u
9.
12.
13.
14.
15.
16.
c  coefficient
d
sec u    sec u tan u  u '
dx
d
18.
csc u     csc u cot u  u '
dx
d
u'
19.
arcsin
u



dx
1 u2
17.
d
20.
 a u    ln a  a u u '
dx
d
sin u    cos u  u ' 21.
dx
d
cos u     sin u  u ' 22.
dx
d
 tan u    sec 2 u  u ' 23.
dx
d
cot u     csc 2 u  u ' 24.
dx
d
u '
arccos u  
dx
1 u2
d
u'
arctan u  
dx
1 u2
d
u '
 arc cot u  
dx
1 u2
d
u'
 arc sec u  
dx
u u2 1
d
u '
 arc csc u  
dx
u u2 1
u & v  expressions in terms of x
33
Homework
Day 1 HW 5.6 p.377
17, 21, 31, 41-61,71,77 all odds
Day 2 HW MMM pgs.205,206
35
Day 2 More Examples. Differentiate:
1. y  arcsin 5x  2

2. y  arctan x 2  5


y 
y 
u u 1
2


5

1  (5 x  2 ) 2
1 u
u
2x

y 

2
1 u
1  ( x 2  5) 2
3. y  csc 1 x 3  2 x  1
 u
u
2

 3x2  2

x 3  2 x  1 ( x 3  2 x  1) 2  1
1
1
1





u
6 
6   1  36
6 
1  x  y  

4. y  cot  
2
2
2
2
2
x
36

x
1

u
6
36

x
x
 
6
1    1
36
36
6
6

36  x 2 36
Practice Problems. Differentiate:
1. y  8 arcsin x  7 arccos x
y  8 
2. y  4 arccos3x  9 y   4 
1
1 x
3
2
1  (3 x  9 )
2
7

1
1 x
 12
2

1
1 x2
1  (3 x  9 ) 2
1
1
1


1  x 
2 
2
3. y  4 tan   y   4  8 2 
x 2 64  x 2
x
8
1    1
64
64
8
1
64
32
 

2
2 64  x
64  x 2
37
Calculus HWQ 1/24
 Find the equation of the tangent line to the graph of the
function at the given point:
x
 
y  arctan   at  2, 
2
 4

1
y    x  2
4 4
38