Implicit Differentiation: Radicals: Find implicitly for .

First we need to convert the radicals to powers, so that we can use the Power Rule.

The problem becomes .

Now we differentiate using the Power 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

. Using the Product and Chain Rules, we get

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.