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Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twilight Zone; we're traveling to another dimension. Once the room stops spinning, we'll move on. Good now? Good. We can handle this.

A function is a special kind of relation where each input has exactly one output. Often we have a rule or a formula that tells us how to reach the input to the output, such as

*y* = *x* + 5.

In such cases we say the function is defined **explicitly**. The dependent variable appears all by itself on one side of the equation, and the other side gives a recipe for making the dependent variable from the independent variable.

We say a function is defined **implicitly** if we have an equation in which the dependent variable doesn't have a side of the equation all to itself. The equation

5*x*^{2} – 4*y* = 7

is an implicit definition of *y*, since it doesn't actually say how to go from *x* to *y*. If we solve this equation for *y* we find the explicit definition

We can say that *y* is "defined implicitly" even if *y* isn't actually a function. For example, the equation

*x*^{2} + *y*^{2} = 1

describes a circle of radius 1. A circle isn't a function, since it fails the vertical line test. However, we would still say that this equation implicitly defines *y*.

For a total headache, we might even say "*y* is an implicitly defined function" even though *y* isn't a function. We probably won't worry about this distinction in class, but we wanted to point it out for the sake of completeness.

### Computing Derivatives Using Implicit Differentiation

We've taken lots of derivatives of explicit functions. In fact, so far we've only taken derivatives of explicit functions. We've had an equation for

*y*(or*f*, or whatever) and we've used our collection of rules to find*f '*.We can also take derivatives of implicit functions. If we have an equation relating

*x*and*y*, we can take derivatives first and solve for*y'*later.We can use either Leibniz or Lagrange notation.

It will be helpful to have a variation on Lagrange notation. The expression (...) means, "take the derivative, with respect to

*x*, of whatever is in the parentheses." It essentially means the same thing as (...)', but with the added precise statement that yes, we are taking the derivative with respect to*x*and not with respect to anything else.For example,

and

Remember, means the same thing as a prime.

(

*f*+*g*)' =*f*' +*g*'We can also say

### Using Leibniz Notation

Some things to remember for implicit differentiation:

- Since
*y*is a function of*x*, any derivative involving*y*must use the chain rule.

- Since
*y*is a function of*x*, taking the derivative of*xy*(or of any other product involving both*x*and*y*) requires the product rule.

- Since
*y*is a function of*x*, taking the derivative of (or any other quotient involving both*x*and*y*) requires the quotient rule.

With these things in mind, we're ready to get cracking.

- Since
### Using Lagrange Notation

Not a fan of Leibniz notation? We can do implicit differentiation with Lagrange notation just as well.

Things to remember for implicit differentiation with Lagrange notation:

*x'*= 1.

- since
*y*is a function of*x*, any derivative involving*y*must use the chain rule.

- since
*y*is a function of*x*, taking the derivative of*xy*(or of any other product involving both*x*and*y*) requires the product rule.

- since
*y*is a function of*x*, taking the derivative of (or any other quotient involving both*x*and*y*) requires the quotient rule.

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