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