## What is Explicit Differentiation?

**Explicit Differentiation** is taking the derivative of an explicit function.

## What is an “Explicit Function”?

In an explicit function,** one variable is defined completely in terms of the other.** This usually means that the independent variable (x) is written explicitly in terms of the dependent variable (y). The general form is: y = f(x). Note that “y” is on one side of the equals sign and “x” is on the other side.

Most of the functions you’re probably familiar with are explicit, like y = x^{2} or y = 2x + 3. When you first start in calculus, practically all of the functions you work with are going to be in this explicit form, and you’ll use the usual rules for differentiation.

## What is “Implicit”?

The opposite of an explicit function is an implicit function, where the variables become a little more muddled. For example, the following equations are implicit:

- x
^{2}+ y^{2}= 1 (x and y are on one side of the equation) - y*e
^{y}= x (two “y”s are on one side of the equation).

## Explicit Differentiation vs. Implicit Differentiation

When you have a function that’s in a form like the above examples, it isn’t possible to use the usual rules of differentiation. When that’s the case, you have two choices:

- Rewrite the equation so that one variable is on each side of the equals sign, then differentiate using the normal rules.
- Use implicit differentiation.

Sometimes, the choice is fairly clear. For example, if you have the implicit function x + y = 2, you can easily rearrange it, using algebra, to become explicit: y = f(x) = -x + 2. In other cases, it might be easier to just use implicit differentiation.

## Example

Let’s say you wanted to differentiate the implicit function x^{4} + 2y^{2} = 4.

**1. Using explicit differentiation:**

Rewrite, using algebra, so that you have one variable on each side of the equals sign:

This would give you two derivatives, one for positive values of y, and one for negative values of y.

**2. Using implicit differentiation:**

Instead of rewriting, you can just go ahead and plug the function in: