
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 $y$ is a differentiable function of $x$ and if $f$ 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 $y$ 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.

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