Derivatives of inverse functions

Inverse Functions Two functions are inverses if The graph of f contains the point (a,b) if and only if the graph of the inverse contains (b,a) Reflect over the line y=x

To Find Inverses Solve the equation for x Interchange x and y Replace y with

let a) sketch the graph b) find the inverse c) sketch the inverse

let d) differentiate both f(x) and f-1(x).

let find the slope of the graph of f(x) at (4, 2) and the slope of the inverse at (2, 4) f) What conclusion can you make about these slopes?

Inverse Functions At right are the graphs of a function f(x) and its inverse f-1(x). Do you see a relationship between the slope of the graph of f at (a,b) and the slope of the graph of the inverse at (b,a)?

conclusion Since slope m = dy/dx, it should make sense that switching x and y (for inverse functions) should produce reciprocal slopes for inverse functions

2) Find the derivative of the inverse of

Derivative of an inverse & image points If (a, b) is a point on f, then (b, a) is a point on , Given x = 5 on the inverse and y = 2, then (5,2) is on the inverse and (2,5) is the image point on the original. Therefore

3) Find the derivative of the inverse

4) Let and let be the inverse of f, then find

Let and

6) Find the derivative of the inverse

 

8)

9) Selected values of g(x) and g’(x) are given in the table. Find

Home Work Derivatives of Inverse functions Worksheet 5-E Use a section header for each of the topics, so there is a clear transition to the audience. Derivatives of Inverse functions Worksheet 5-E