5.6 Inverse Trigonometric Functions: Differentiation
Definitions of inverse trig functions – Note the restricted domains of the original trigonometric functions
so that the inverse is also a function.
Function
y  arcsin x iff sin y  x
Domain
1  x  1
y  arccos x iff cos y  x
y  arctan x iff tan y  x
1  x  1
  x  
y  arc cot x iff cot y  x
y  arc sec x iff sec y  x
  x  
y  arc csc x iff csc y  x
x 1
x 1
Examples: Evaluate the expression without using a calculator.
1. arcsin 0
2. arccos1

3. arc cot  3



4. tan  arccos

2

2 
Range


 y

2
2
0 y 


  y
2
2
0 y 
0 y  , y 


2
 y

2

2
, y0


5

13 

 3 


5. cos  arcsin
6. sec arctan    
5

Derivatives of Inverse Trig Functions – Let u be a differentiable function of x.
d
u'
arcsin u  
dx
1 u2
d
u'
arctan u  
dx
1 u2
d
u'
 arc sec u  
dx
u u2 1
Examples: Find the derivative of the function.
1. f  t   arcsin t 2
2. f  x   arc sec 2 x
d
u '
arccos u  
dx
1 u2
d
u '
 arc cot u  
dx
1 u2
d
u '
 arc csc u  
dx
u u2 1
3. f  x   arctan x
4. h  x   x 2 arctan 5x
5. f  x   arcsin x  arccos x