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AP Calculus Review
Inverse Trigonometric Functions: Differentiation

Find the derivative of inverse trigonometric functions
None of the six basic trigonometric functions have inverses. But, if the domain is restricted, each of the
trigonometric functions can be defined in such a way that they are all one-to-one. Hyperbolic functions occur in
the solutions of some important differential equations and are generally studied I n physics. More simply,
hyperbolic trigonometric functions are rational functions of exponentials.
Definitions of the Inverse Trigonometric Functions:
Function
Domain
y  arcsin x if and only if sin y  x
1  x  1
y  arccos x if and only if cosy  x
1  x  1
y  arctan x if and only if tany  x
  x  
y  arc cot x if and only if coty  x
  x  
Range


y  arc sec x if and only if sec y  x
x 1

y  arc csc x if and only if csc y  x
x 1

Graphs of the Inverse Trigonometric Functions
Derivatives of the Inverse Trigonometric Functions

 y

2
2
0 y 

 y

2
2
0 y 

2

2
 y
 y

2

2
,y

2
,y0
Differentiate each of the following functions.
y  x 2 arcsin x
y  x arcsin x  1  x 2
y  arc sec x  arc csc x

y  arcsin  x3 

y
1  arctan x
2  3arctan x
4
1
y  arc cot    arc tan x
 x
 
y  arctan e x
2
Determine the slope of the tangent line to the graph of y  arctan  ln x  at x  e.
Assignment: Worksheet (457) #1-36