Implicit Differentiation

Suggested Prerequesites: The definition of the derivative, The chain rule

There are two ways to define functions, implicitly and explicitly. Most of the equations we have dealt with have been explicit equations, such as y = 2x + 3, so that we can write y = f(x) where f(x) = 2x + 3. But the equation 2x - y = -3 describes the same function. This second equation is an implicit definition of y as a function of x.

Not all implicit equations can be restated explicitly. For example, the implicit equation x2 + y2 = 9 needs two explicit equations, and , which are the top and bottom halves of a cricle respecively, to define it completely. And xy = sin(y) + x2y2 may have an explicit definition, but the process of finding it would be so tedious that we'd rather not need to do so.

Surprisingly, we can take the derivative of implicit functions just as we take the derivative of explicit functions. We simply take the derivative of each size of the equation, remembering to treat the dependent variable as a function of the dependent variable, apply the rules of differentiation, and solve for the derivative. Returning to our original example:

To check, we take the derivative explicitly:

Some examples:

  1. Remember to apply the chain rule -- y is a function of x!

    It is acceptable, and often unavoidable to define y' in terms of y. After all, if we could solve for y explicitly, we wouldn't need implicit differentiation.

  2. It is important that we remember the chain rule and the product rule for terms such as xy:

In implicit differentiation, and in differential calculus in general, the chain rule is the most impostant thing to remember!

Exercises:

    Differentiate the following equations explicity. Assume that y is a function of x. Solve for y'.

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page
jjnichol@mit.edu
Last Modified 23 June 1997