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Derivatives of Inverse Functions

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Derivatives of Inverse Functions

AP Calculus

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.

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

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

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

(a,b)

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

(a,b)

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!!!

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

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

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

Theorem:

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

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

- 10/18/10
- Assignment: Worksheet 91