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.

Calculus I: Basic Rules For The Derivative

After mastering the previous lesson, Calculus I: The Derivative, you will be pleased to know that we NEVER use that again to take a derivative. With a more complicated problem, that technique becomes extremely tedious, and excessively algebraic. It is something you need to know, because you will most likely be tested on it, but we can now move on to all the great shortcuts!

A good place to start:

If you have a constant, what is its derivative?

Find the derivative of:

The derivative of any constant, let’s call it ‘a’, is 0.


These all mean the exact same thing.

What is the derivative of just a simple x?


Again, these all mean the exact same thing!

What about a constant times x? For example, 6x?

You are allowed to take out the 6, and move it off to the side. Then you take the derivative of what is left, and multiply it by the 6 afterwards.

Now, we can generalize this for any constant, ‘a’.

These are the most basic things that we can take the derivative of. Now would be a good time to show you what to do with a slightly more complicated function, such as .

The rule you need to know is this:


(This also works for subtraction!)

Basically, this means that if you have a function that has two parts, to take the derivative of the function, you can just take the derivative of each part separately. This only works for addition and subtraction!

What is the derivative of the following functions?


The derivative of 3x is just 3, and the derivative of 6 is 0. This means:

If you need more practice with these, just make up some simple linear functions, and find their derivative!
You could try:

Okay, so how could I make this any more complicated and foreign to you? How about by throwing in exponents!? I think that would be fun =) So, what do we do with an exponent?

To be honest, this is going to seem really weird at first, but just go with it. Believe it. How do you take the derivative of ?


The first step is to take the exponent (the 2), and put it in front of the x. Just move it on down, and multiply it. Next, you have to subtract 1 from the original exponent. If you had x to the fourth,  it would become x cubed, 4x³. Okay, so let’s get back to the problem at hand.


So we bring down the exponent, the 2, and we subtract 1 from the exponent. That is the answer.

We subtract 1 from the exponent, making it a 1. The answer can be simplified.

Let’s try another.


What do we do now with the coefficient? If you remember from before, you put the coefficient, the 2, off to the side and take the derivative without it. Then you multiply it back in afterwards. So, pretend the 2 isn’t there. What is the derivative of x³?
So, leave the two off to the side. Then bring down the 3 and subtract 1 from the exponent. Let’s do it!

Two more, slightly harder ones?

First, you need to recognize that when you have an x in the denominator (the bottom), it is the EXACT same as a negative exponent. The actual number will remain the same, it just becomes a negative when you move it to the numerator (the top). So, to take a derivative with an x in the denominator, you should move it to the numerator.

Something to be EXTRA careful of, is to SUBTRACT 1 from the exponent. It is negative, so when you subtract 1 it becomes more negative. This means that the negative number gets bigger. So if you started with a -2, you would subtract 1 and it would be a -3. That is an extremely common mistake. You’ve been warned!

There is one last step to this problem. Notice that the original problem had the x in the denominator. As we have it now, we are using a negative exponent. You always have the option to use either a negative exponent or to put it in the denominator. So, how are you supposed to leave your answer? This is a big pet peeve of mine! You should always leave the answer just like the problem was originally stated. If the original problem used fractions, put it in the denominator. If the problem started with a negative exponent, make it a negative exponent. So, we change it to this:


This is the way you should leave your answer!

What about fractional exponents? This can be difficult at first.

Can you follow what I did? I left the coefficient off to the side, and I brought down the exponent. Then, you subtract 1 from ½. That becomes -½. Then, to simplify, the 2 and the ½ cancel!

Again, I’m going to make it more difficult on you now. I’m sorry! Here are several facts that you must memorize. I’m very sorry, but I’ll try to make it as easy as possible for you.

Those 6 are pretty important. It would behoove you to memorize them =)  The first three are the most important, just so you know.Here is my trick: The co’s (cosine, cotangent, cosecant), all become negative when you take the derivative. The next thing to notice is that for sec and tan and csc and cot, they are paired up; tan sticks with sec and cot sticks with csc. The regular ones stick together, and the co’s stick together. That is my best advice on how to memorize them. I know it’s not too much, but hopefully it will be of some help. The real best way to memorize them is to practice!!!!

Since it is adding (or subtracting) we can take the derivative of each part separately. From above, we know that the derivative of sin is cos. So, let’s apply that. Also, remember to mentally keep coefficients off to the side as you take the derivative.

See how it works? These functions can begin to get very complicated, very quickly. Just try a few, and you will get the hang of it for sure. There are a lot of small rules that you just have to learn, unfortunately. Soon, they will be second nature to you.

One last problem-

If you have any questions or comments, please post below! Good luck!

Be sure to check out the next lesson on the Product Rule.

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