BC 1
Inverse Functions and Their Derivatives
(1)
Sketch ƒ(x) = sin 1 x .
Name:
(2)
Find exact values.
  3
1 
sin 1 
sin 1  

2 
 2 
sin 1 1
Domain:
 2
sin 1 

 2 
Range:
(3)
Sketch g(x) = cos1 x .
(4)
Find exact values.
1
cos 1
cos1 1/ 2
 3
cos1 

 2 
 2
cos1 

 2 
Domain:
Range:
(5)
Sketch h(x) = tan 1 x .
(6)
Find exact values.
tan1  3
 3
tan 1 

 3 
tan 1 1
tan 1 0
 
Domain:
Range:

Inv. Functions p.1
SP14
(7)
Find exact values of each of the following. (Hint: Draw triangles.)

5 
(b) sin tan 1 3
(a) cossin 1 

13
(8)
Assume a, b, and c are in the appropriate domains for each problem. Find each of the
following.
1
1
(a) sin cos a
(b) tan cos a




1

(c) cos sin b
(9)




1

(d) tan sin b
Since eln( x )  x , we can differentiate both sides with respect to x to get:
d ln( x )
d
e   ( x)

dx
dx
d
 eln( x )   ln( x)   1
dx
d
 x   ln( x)   1
dx
d
1
  ln( x)  
dx
x
Find the derivatives of each function:
(a) f ( x)  ln( x 2  3x)
(b) f ( x)  x ln(sin( x))
Inv. Functions p.2
SP14
(10)
We now want to find the derivatives of the trig inverse functions.
Let y  sin 1 ( x) . We wish to find
sin( y )  x .
dy
. To this end, we solve for x, giving
dx
Differentiate both sides with respect to x, then solve for
So,
dy
.
dx
dy
1
.

dx cos( y )
Using that y  sin 1 ( x) , simplify the expression for
dy
. See problem (8).
dx
So, if f ( x)  sin 1 ( x) , then f ( x) 
Find the derivatives of each function:
(a) f ( x)  sin 1 ( x 2  3x)
1
1  x2
(b) f ( x)  x sin 1 ( x)
Inv. Functions p.3
SP14
(11)
Using the same process as in problem (10), find the derivatives of y  cos 1 ( x) and
y  tan 1 ( x) .
1. If f ( x)  cos 1 ( x) , then f ( x) 
1
1  x2
1
2. If f ( x)  tan 1 ( x) , then f ( x) 
1  x2
Inv. Functions p.4
SP14