In this section we differentiate equations that contain more than one variable on one side.

Review of the chain rule

Implicit differentiation is really just an application of the chain rule. So recall:

Of particular use in this section is the following. If is a differentiable function of and if is a differentiable function, then

Implicit differentiation

The functions we’ve been dealing with so far have been defined explicitly in terms of the independent variable. For example: However, this is not always necessary or even possible to do. Sometimes we choose to or we have to define a function implicitly . In this case, the dependent variable is not stated explicitly in terms of an independent variable. Some examples are: Your inclination might be simply to solve each of these equations for and go merrily on your way. However, this can be difficult or even impossible to do. Since we are often faced with a problem of computing derivatives of such functions, we need a method that will enable us to compute derivatives of implicitly defined functions.

We’ll start with a basic example.

Let’s take a different approach, namely let’s use implicit differentiation.

Let’s see another illustrative example:

You might think that the step in which we solve for could sometimes be difficult. In fact, this never happens. All occurrences arise from applying the chain rule, and whenever the chain rule is used it deposits a single multiplied by some other expression. Hence our expression is linear in , it will always be possible to group the terms containing together and factor out the , just as in the previous examples.

One more last example: