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Implicit Functions

We now turn to the topic of implicit differentiation. We say that y is an implicit function of x if we are given an equation

σ(x, y) = τ(x, y)

which determines y as a function of x. An example is x + xy = 2y. Implicit differentiation is a way of finding the derivative of y without actually solving for y as a function of x. Assume that dy/dx exists. The method has two steps:

Step 1

Differentiate both sides of the equation σ(x, y) = τ(x, y) to get a new equation



The Chain Rule is often used in this step.

Step 2

Solve the new Equation 1 for dy/dx. The answer will usually involve both x and y.

In each of the examples below, we assume that dy/dx exists and use implicit differentiation to find the value of dy/dx.


In Example 1, we found dy/dx by three different methods.

(a)    Implicit differentiation. We get dy/dx in terms of both x and y.

(b)    Solve for y as a function of x and differentiate directly. This gives dy/dx in terms of x only.

(c)     Solve for x as a function of y, find dx/dy directly, and use the Inverse Function Rule. This method gives dy/dx in terms of y only.

Last Update: 2006-11-25