Implicit Differentiation: Product of x and y: Find implicitly for .

We shall use the Power Rule and the Product Rule:

. Solving for , we get

.

In this case (but not usually), we can solve the original problem for y. By doing this, we can find explicitly.

Solving the original problem for y, we get

Now we use the Quotient Rule:

In order to compare this result with the first one, we need to replace y in that first result:

.

Upon simplification, this reduces to the second result. In other words, we get the same result by these two methods. Note that the second method frequently is not available - when the original equation can not be solved for y explicitly.