 ## 2.11 Implicit Differentiation

### Summary

Not all curves in the plane (given by an equation in x and y) are the graphs of functions. However, if it is assumed that for a piece of the curve, the equation determines implicitly a differentiable function y = f(x), then it is possible to find the derivative of f without explicitly finding a formula for y in terms of x. The method is called that of Implicit Differentiation.

By the end of your studying, you should know:

• How to apply the method of Implicit Differentiation.
• How to find the derivative of the inverse of a function.

On-screen applet instructions: The button at the very bottom gives the interval over which you are tracing where it is possible to define y as a function of x. Use this button only to check your work after you have tried to find the interval on your own.

### Examples Find y' by implicit differentiation, where xy = cot(xy). Find the tangent line to the ellipse at the point  An interesting curve first studied by Nicomedes around 200 B.C. is the conchoid, which has the equation x2y2 = (x + 1)2 (4 – x2). Use implicit differentiation to find a tangent line to this curve at the point (–1, 0).

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.11 Implicit Differentiation (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Implicit Differentiation
Visual Calculus
Kouba

#### Interesting Application Nothing yet has been found. Any ideas?

Software requirements: For best results viewing and interacting with this page, get the free software listed here.

Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel