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Calculus I: The Product Rule

In this lesson, I’ll be using rules that you can find in my previous lesson on basic derivative rules.

So, what is the product rule, and when do we use it? It is very, very important to be able to recognize when you need the product rule. You NEED to use the product rule whenever you have two functions of x multiplied together. This can take many forms.

For example:

Whenever there are two functions of x, multiplied together, you need  to use it. Got it? Always be on the lookout from now on!

What is the product rule?

 
(This just means two functions of x multiplied together, like the examples above)

The product rule says:

“y prime equals the derivative of the first times the second, plus the first times the derivative of the second!”
This little mantra will help you remember the formula. “the first” is referring to the first function, f(x), and “the second” is referring to the second function, g(x).

How, exactly, is it used?

The best way for me to demonstrate this is with an example. We should start with something very simple.

All I did was use the forumla! Recognize that f(x) is (x + 5) and that g(x) is (3x – 2). Keep saying it to yourself (out loud if you wish) as you are writing down the derivative. “y prime equals the derivative of the first times the second, plus the first times the derivative of the second!”

For the next example, let’s do something a little bit harder.

We first need to recognize that the product rule is needed. As you can see, there are two separate functions of x being multiplied together. There is just an x, and there is sinx. Then, we just follow our formula! Simple as that.

Now, what do you do if there are three functions of x being multiplied together? This gets a bit tricky, but I’m sure you will catch on to it. To take the derivative, you must treat it as if it is only two functions. In the example below, pretend that the first function, f(x) is the x. Then, the second function has to be everything that is left. g(x) will be the . Next, you must do the product rule like we always do!

Here is the trick: What is g ‘(x)? Have you guessed it? You must do another product rule, inside of the first product rule!

I’ll break this down for you nice and slowly.

It’s time for you to go practice now. If you have any questions, you know how to contact me! Come back for my next lesson on the quotient rule.