Derivatives of Inverse Functions

1. Derivatives of Inverse Functions AP Calculus

2. 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.

3. 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

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

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

6. (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

7. Find the derivative of the inverse by implicit differentiation ( without solving for f -1 (x) ) Remember : f -1 (x) = f (y) ; therefore, find

8. (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!!!

9. ILLUSTRATION: Find the derivative of f -1at (16,4) a) Find the Inverse. b) Use the formula.

10. Find the derivative of the Inverse at the given point, (b,a). EX: Theorem:

11. Inverse Functions If S(x) = f -1 (x), then S / (3) = If S(x) = f -1 (x), then S / (10) =

12. Last Update • 10/18/10 • Assignment: Worksheet 91