Download Derivative of the Arcsine

3.8
Derivatives of Inverse Trigonometric Functions
S
Quick Review
In Exercises 1-5, give the domain and range of the function,
and evaluate the function at x 1.
1. y  sin 1 x
2. y  cos 1 x
3. y  tan 1 x
4. y  sec 1 x
5. y  tan  tan 1 x 
Slide
3- 2
Quick Review
In Exercises 6-10, find the inverse of the given function.
6. y  3 x  8
7. y  3 x  5
8
x
3x  2
y
x
8. y 
9.
x
10. y  arctan  
3
Slide
3- 3
Quick Review Solutions
In Exercises 1-5, give the domain and range of the function,
and evaluate the function at x 1.
1. y  sin 1 x
2. y  cos 1 x
3. y  tan 1 x
4. y  sec 1 x
5. y  tan  tan 1 x 

  
Domain: 1,1 Range: - ,  At 1:
2
 2 2
Domain: 1,1 Range: 0,   At 1:0

  
Domain:All Reals Range: - ,  At 1:
4
 2 2
Domain:  ,  1 ∪ 1,  
    
Range: 0,  ∪  ,   At 1:0
 2 2 
Domain:All Reals Range:All Reals At 1:1
Slide
3- 4
Quick Review Solutions
In Exercises 6-10, find the inverse of the given function.
x8
3
6. y  3 x  8
f 1  x  
7. y  3 x  5
f 1  x   x 3  5
8. y 
8
x
3x  2
y
x
f 1  x  
8
x
9.
f 1  x  
2
3 x
x
10. y  arctan  
3
f 1  x   3tan x, 

2
Slide
3- 5
x

2
What you’ll learn about
S Derivatives of Inverse Functions
S Derivatives of the Arcsine
S Derivatives of the Arctangent
S Derivatives of the Arcsecant
S Derivatives of the Other Three
… and why
The relationship between the graph of a function and its inverse
allows us to see the relationship between their derivatives.
Slide
3- 6
Derivatives of Inverse
Functions
dy
dx
is differentiable
If f is differentiable at every point of an interval I and
is never zero on I , then f has an inverse and f 1

at every point on the interval f I .
Slide
3- 7
Derivative of the Arcsine
If u is a differentiable function of x with u  1, we apply the
Chain Rule to get
d
1
du
1
sin u 
, u  1.
2
dx
1 u dx
Slide
3- 8
Let f(x) = sin x and g(x) = sin-1 x to verify the
formula for the derivative of sin-1 x.
Example Derivative of the
Arcsine
dy
If y  sin 8x , find
.
dx
1
2
Slide
3- 10
Example Derivative of the
Arcsine
If y  sin 1 (1 t), find
dy
.
dx
Slide
3- 11
Derivative of the Arctangent
The derivative is defined for all real numbers.
If u is a differentiable function of x, we apply the
Chain Rule to get
d
1 du
1
tan u 
.
2
dx
1 u dx
Slide
3- 12
dy
Determine
dx.
y = tan-1 (4x)
dy
Determine
dx.
y = x tan-1x
Derivative of the Arcsecant
If u is a differentiable function of x with u  1, we have the
formula
d
1
du
1
sec u 
, u  1.
dx
u u 2  1 dx
Slide
3- 15
Example Derivative of the
Arcsecant
dy
Given y  sec  3x  4  , find
.
dx
1
Slide
3- 16
A particle moves along the x – axis so that
its position at any time t ≥ 0 is given by
x(t). Find the velocity at the indicated
value of t.


t
x(t)  sin 1   , t  4
 4
Assignment 3.8.1
page 170,
# 3 – 11 odds
Inverse Function – Inverse
Cofunction Identities
1
cos x 
1
cot x 
1
csc x 

2

2

2
1
 sin x
1
 tan x
 sec 1 x
Slide
3- 19
dy
Determine
if y  cos1 x.
dx
Derivatives of Inverse Trig
Functions
Function arcsin x arccos x arctan x arcsec x
1
1
Derivative
1 x
2
1  x2
1
2
1 x
1
x
x2 1
Example Derivative of the
Arccotangent
1
Find the derivative of y  cot x .
Slide
3- 22
2
Calculator Conversion
Identities
1
sec x  cos  
x
1
1
cot x 
1

2
 tan 1 x
1
csc x  sin  
x
1
1
Slide
3- 23
Determine the derivative of y with respect to
the variable.
1 
1
y  cos  
 x
Determine the derivative of y with respect to
the variable.
1
y  sec 5s
Determine the derivative of y with respect to
the variable.
y  csc
1
x
2
Determine the derivative of y with respect to
the variable.
1
y  s  1  sec s
2
Find an equation for the tangent to the graph
of y at the indicated point.
1
y  tan x, x  2
Find an equation for the tangent to the graph
of y at the indicated point.
1 
x
y  cos   , x  5
 4
Let f(x) = cos x + 3x
Show that f(x) has a differentiable inverse.
Let f(x) = cos x + 3x
Determine f(0) and f ’(0).
Let f(x) = cos x + 3x
Determine f-1(1) and f-1(1).
y=
-1
cot
x
Determine the right end behavior model.
y=
-1
cot
x
Determine the left end behavior model.
y=
-1
cot
x
Does the function have any horizontal tangents?
Assignment 3.8.2
pages 170 – 171,
# 1, 13 – 29 odds, 32 and 41 – 45 odds