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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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


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.



Inverse functions
Inverse Functions (b,a).

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

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


Last update
Last Update (b,a).

  • 10/18/10

  • Assignment: Worksheet 91


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