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).