derivatives of inverse functions
DESCRIPTION
Derivatives of Inverse Functions. AP Calculus. Inverses. Existence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further, Domain of f = Range of f -1 Range of f = Domain of f -1. - PowerPoint PPT PresentationTRANSCRIPT
Derivatives of Inverse Functions
AP Calculus
InversesExistence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further,
Domain of f = Range of f -1
Range of f = Domain of f -1
One-to One Functions: A function f(x) is one-to one (on its domain D) if for every x there exists only one y
and for every y there exists only one x
Horizontal line test.
REVIEW: Inverse Functions
(a,b)
(b,a)
If f(x) is a function and ( x, y) is a point on f(x) , then the inverse f -1(x) contains the point ( y, x)
Theorem:
f and g are inverses iff
f(g(x)) = g(f(x)) = x
To find f -1(x)
Reverse the x and y and resolve for y.
31
xyx
Restricting the Domain:If a function is not one-to-one the domain can be restricted to portions that are one-to-one.
x
y
Restricting the Domain:If a function is not one-to-one the domain can be restricted to portions that are one-to-one.
x
y
3( ) 5 1f x x x
Derivative of the Inverse
Derivative of an Inverse Function:
Given f is a differentiable one-to-one function and f -1 is the inverse of f . If b belongs to the domain of f -1 and
f /(f(x)) ≠ 0 , then f -1(b) exists and
(a,b)
(b,a)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
1
/ 1
1( )( )
f bf f b
( ) 2 sin( )f x x x
Find the derivative of the inverse by implicit differentiation( without solving for f -1 (x) )
Remember : f -1 (x) = f (y) ; therefore,
find dxdy
Derivative of the Inverse
Derivative of an Inverse Function:
If is the derivative of f,
Then is the derivative of f -1(b)
x a
dydx
1
x a
dydx
(a,b)
(b,a)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
CAUTION:
Pay attention to the plug in value!!!
ILLUSTRATION:2
1
( )
( )
f x x x o
f x x
Find the derivative of f -1 at (16,4)
2( )f x x
1( )f x x
a) Find the Inverse. b) Use the formula.
EX:Find the derivative of the Inverse at the given point, (b,a).
3( ) 7 6f x x at b
Theorem: 1 1( )( )
f bf a
Inverse Functions
x f f /
10 3 23 10 4
If S(x) = f -1 (x), then S / (3) =
If S(x) = f -1 (x), then S / (10) =
Last Update
• 10/18/10
• Assignment: Worksheet 91