Derivatives of inverse functions
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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.

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Derivatives of Inverse Functions

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Derivatives of inverse functions

Derivatives of Inverse Functions

AP Calculus


Inverses

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

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

REVIEW: Inverse Functions

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)

To find f -1(x)

Reverse the x and y and resolve for y.

(a,b)

(b,a)

Theorem:

f and g are inverses iff

f(g(x)) = g(f(x)) = x


Restricting the domain

Restricting the Domain:

If a function is not one-to-one the domain can be restricted to portions that are one-to-one.


Restricting the domain1

Restricting the Domain:

If a function is not one-to-one the domain can be restricted to portions that are one-to-one.


Derivative of the inverse

(a,b)

Derivative of the Inverse

The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.

(b,a)

Derivative of an Inverse Function:

Given f is a differentiable one-to-one function and f -1is the inverse of f . If b belongs to the domain of f -1and

f /(f(x)) ≠ 0 , then f -1(b) exists and


Derivatives of inverse functions

Find the derivative of the inverse by implicit differentiation

( without solving for f -1 (x) )

Remember : f -1 (x) = f (y) ; therefore,

find


Derivative of the inverse1

(a,b)

Derivative of the Inverse

The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.

(b,a)

Derivative of an Inverse Function:

If is the derivative of f,

Then is the derivative of f -1(b)

CAUTION:

Pay attention to the plug in value!!!


Illustration

ILLUSTRATION:

Find the derivative of f -1at (16,4)

a) Find the Inverse.b) Use the formula.


Derivatives of inverse functions

Find the derivative of the Inverse at the given point, (b,a).

EX:

Theorem:


Inverse functions

Inverse Functions

If S(x) = f -1 (x), then S / (3) =

If S(x) = f -1 (x), then S / (10) =


Last update

Last Update

  • 10/18/10

  • Assignment: Worksheet 91


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