Arctan & Arcsin Essay

Derivatives of Inverse Trigonometric Functions
Prior Knowledge
The vertical test to determine if a graph is the graph of a function.
The horizontal line test to determine if the graph of a function is the graph of a one-to-one function.
Recall how the inverse of a one-to-one function is defined.
Right-triangle trigonometry.
Pythagorean identities.
Graphs of trigonometric functions.
Derivatives of trigonometric functions.
The chain rule.

Definition of y = f x = arctan x = tan-1 x

First note that the graph of y = tan x does not pass the horizontal line test. Thus, it is necessary to restrict the domain of tangent to ensure that the graph passes the horizontal line test. As the graph below shows, this is accomplished by restricting the domain to p p
- 2 , 2 .
4

p

2

-

3p

-

2

2

p

p

3p

5p

2

2

2

2

-2

2

-2

-

p
2

-4

p p
- 2 , 2 

Graph of y = arctan x = tan-1 x
Domain
-¶, ¶

Graph of y = tan x

Range -¶, ¶

Domain to

- 2 , 2  p Range

p

The following is valid: y = arctan x ó tan y = x, as long as x in -¶, ¶ and y in - 2 , 2 

Computing

„

„x

p

p

arctan x

The key components are: (1) The definition of arctangent, (2) the chain rule, (3) the Pythagorean identity 1 + tan2 q = sec2 q. p p
Setp 1 Set u = ux = arctan x ï tan u = x, for x in -¶, ¶ and u in - ,
.
2 2
Setp 2 Take the derivative of both sides of the last equation, and solve for
„
„x

tan u =

„
„x

„

x ï sec2 u

„x

Setp 3 Use a Pythagorean identity:

Step 4 Rewrite in terms of x:

„
„x

u=1 ï

„
„x

„
„x

1

u=

=

2

sec u

arctan x =

1
1 + x2

u=

1 sec2 u
1

1 + tan2 u

„

„x

u

2

M140_DerivOfInverseTrig_PF.nb

Definition of y = f x = arcsin x = sin-1 x

First note that the graph of y = sin x does not pass the horizontal line test. Thus, it is necessary to restrict the domain of tangent to ensure that the graph passes the horizontal line test. As the graph below shows, this is accomplished by restricting the domain to

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