# Derivatives of Inverse Functions ln(x) and e^x

- Topic:
- Functions

This applet can be used to induce the relationship between the derivative of a function at a point and the derivative of the inverse of the function at the corresponding point.

1. How are the functions y=e^x and y=ln(x) related?
Use the slider tool to answer the following questions.
2. What is the slope of y=e^x at the point (0, 1)? What is the slope of y=ln(x) at the point (1, 0)? How are the slopes related?
3. What is the slope of y=e^x at the point (1, e)? What is the slope of y=ln(x) at the point (e, 1)? How are the slopes related?
4. What is the slope of y=e^x at the point (2, e^2)? What is the slope of y=ln(x) at the point (e^2, 2)? How are the slopes related?
5. Based on your observations from questions 2 through 4, make a conjecture about the derivative of a function f at a point (a, b) and the derivative of the inverse of f, g, at the corresponding point (b, a).