Download Section 2.3 Derivatives of Power Functions and Polynomials

In this section, we will introduce the
inverse trigonometric functions and
construct their derivative formulas.
A function is called one-to-one if whenever f ( x1 ) = f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
A function is called one-to-one if whenever f ( x1 ) = f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
This function is not one-to-one.
A function is called one-to-one if whenever f ( x1 ) = f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
Significance:
Only one-to-one functions have inverse functions.
Below is shown the graph of y = f ( x ) = sin ( x )
This function is not one-to-one and so has no inverse function.
Below is shown the graph of y = f ( x ) = sin ( x )
Consider restricting the domain
of the sine function to: [- p2 , p2 ]
This is the function in blue
shown to the left.
This function has an inverse.
The function f ( x ) = arcsin ( x ) is the inverse of the sine
function with restricted domain [- p2 , p2 ].
That is, the arcsin(x) is the angle θ in the interval [- p2 ,
with sin (q ) = x .
p
2
]
Below is shown the graph of y = f ( x ) = cos ( x )
This function is not one-to-one and so has no inverse function.
Below is shown the graph of y = f ( x ) = cos ( x )
Consider restricting the domain of
the cosine function to: [ 0, p ]
This is the function in blue shown
to the left.
This function has an inverse.
The function f ( x ) = arccos ( x ) is the inverse of the cosine
function with restricted domain [ 0, p ] .
That is, the arccos(x) is the angle θ in the interval [ 0, p ]
with cos (q ) = x.
Below is shown the graph of y = f ( x ) = tan ( x )
This function is not one-to-one and so has no inverse function.
Below is shown the graph of y = f ( x ) = tan ( x )
Consider restricting the domain of
the tangent function to: ( - p2 , p2 )
This is the function in blue shown
to the left.
This function has an inverse function.
The function f ( x ) = arctan ( x ) is the inverse of the tangent
function with restricted domain (- p2 , p2 ).
That is, the arctan(x) is the angle θ in the interval (- p2 ,
with tan (q ) = x .
p
2
)
Use the definitions of the section to find the exact
value of tan ( arcsin ( 97 )) .
Use the definitions of the section to find the exact
value of csc ( arccos (- 114 )) .
Use the definitions of the section to find the exact
(
( ))
value of sec arctan x 2 .
Use the definitions of the section to find the exact
(
value of cos arcsin
( x )).
The following are true:
f ( x ) = arcsin x
Þ f ¢( x) =
f ( x ) = arccos x
Þ f ¢( x) =
f ( x ) = arctan x
Þ f ¢( x) =
1
1- x 2
-1
1- x 2
1
1+ x 2
The following are true:
f ( x ) = arcsec x
Þ f ¢( x) =
f ( x ) = arccsc x
Þ f ¢( x) =
f ( x ) = arccot x
Þ f ¢( x) =
1
x
x 2 -1
-1
x
x 2 -1
-1
1+ x 2
( )
Find the derivative of the function f ( x ) = arctan x 2 .
( )
Find the derivative of the function f ( x ) = arcsin e3x .
Find the derivative of the function f ( x ) = arccos ( ln x ).