# Understand the Derivative Product Rule

As you may know, the operation that apply to numbers do not necessary apply for derivatives too. A good example is the product operation. The **derivative product rule **is a little bit more special and hard to understand and prove.

The **product rule for derivatives **is used when we have to differentiate two functions that are being multiplied together.

Basically, the **product rule derivatives **are suffering the following operation:

In words, the **product rule definition** says that: the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

## Let's see some Product Rule Examples

In order to get used with the rule, we provide you a series of **product rule examples** and **product rule problems**.

1.) Find f'(x):

Solution:

The first term is:

The second term is:

Then, we add them together and obtain our derivative:

2.) Differentiate:

Solution:

3.) The third example constitute in **the zero product rule**: any derivative multiplied with zero results zero.

4.) In the fourth example we will present the **triple product rule**. Basically, it is the **product rule with 3 terms**.

After we saw the **product rule with three terms**, it's time we have a look at the **product rule proof**.

**Proof Of Product Rule**

The **proof of product rule** is not so easy to make, but if you pay attention on the demonstration you will surely understand it.

The definition of derivative:

The key is to subtract and add a term:

By definition,

and

Also

and

In conclusion, we reach the **product rule formula**:

Now that the **proof of the product rule** is complete, it's time we find out a few more facts about our subject.

For radicals, the **product rule derivative** is suffering the same transformations, so the **product rule for radicals** is basically the same.

The equality in the formula is true in both sides, which means that the **reverse product rule** is also valid.

Engineer
San Francisco, USA
"If you're at school or you just deal with mathematics, you need to use Studygeek.org. This thing is really helpful." |
Math Teacher
New-York, USA
"I will recommend Studygeek to students, who have some chalenges in mathematics. This Site has bunch of great lessons and examples. " |
Student, Designer
Philadelphia, USA
" I'm a geek, and I love this website. It really helped me during my math classes. Check it out) " |
Bookkeeper
Vancuver, Canada
"I use Studygeek.org a lot on a daily basis, helping my son with his geometry classes. Also, it has very cool math solver, which makes study process pretty fun" |