This section we are going to talk about the product rule. This section we are going to talk about the product rule. If we are looking for the product rules according to the formula, If we are looking for the product rules, according to the formula, D stands for the derivative of two separate functions, D stands for the derivative of two separate functions, f of x times s of x. f of x times s of x. And you are going to find the derivative of one time of each function with a plus addition in between.
And you are going to find the derivative of one time of each function with a plus addition in between. We give you example We give you an example. 1 from the guided example and the example 5 on the guided example.
1 from the guided example and the example 5 on the guided example. Let's start the first example, Let's start the first example. example 1. Example 1. You can easily see example 1, You can easily see example 1. y is equal to y is equal to 2x plus 1 times 2x plus 1 times 3x plus 5. 3x plus 5. Basically, Basically, you can just treat it as f of x. you can just treat it as f of x and s of x. And the s.
of x. And I normally like to rewrite the question y prime. And I normally like to rewrite the question y prime.
It's equal to, It's equal to, I'm going to write down the original question twice. I'm going to write down the original question twice. 2x plus 1 times 3x plus 5. And make sure go by the formula.
And make sure to go by the formula. Plus 2x plus 1, Plus 2x plus 1, 3x. 3x plus 5. And we are going to use, And we are going to use make sure different color marker and make sure, a different color marker. And I'm going to find the derivative of I'm going to find the derivative of the second one and find the derivative of the first one.
The second one and find the derivative of the first one go by the formula Go by the formula. Now let's solve the question. Now let's solve the question.
It's equal to It's equal to 2x plus 3. 2x plus 3. You don't touch it because you are trying to find the derivative of the 3x plus 5. You don't touch it because you are trying to find the derivative of the 3x plus 5. Derivative of the 3x is... Derivative of the 3x is... is equal to 3. is equal to 3. Plus 5 is a constant.
Plus 5 is a constant, it doesn't have the x, It doesn't have the x, so there's no changes, so there's no changes. that means it's 0. That means 0. Plus. Plus, derivative of the Derivative of the 2x is equal to 2. 2x is equal to 2. Plus 1 is a constant. Plus 1 is a constant, it doesn't have the x, It doesn't have the x.
it goes away, It goes away. equal to nothing. Equal to nothing times bring down.
Times, bring it down. 3x plus 5, 3x plus 5. Don't forget. okay. And you just completed the derivative.
And you just completed the derivative. The following steps you have to do is to distribute and combine like terms. The following steps you have to do is to distribute and combine like terms. That's all.
That's all. Let's continue to finish that. Let's continue to finish that.
2x times 3, 2x times 3, 6x. 6x. Plus 3 times 1, Plus 3 times 1, 3. 3. Plus Plus 2 times 2 times 3x, 3x, 6x.
6x. plus plus 2 times 2 times 5, it's 10. 5 is 10. Now let's combine the like terms. Now let's combine the like terms.
Equals. Equals. So we can see easily, So we can see easily 6x plus 6x, 6x plus 6x, 12x in all, 12x in all, plus plus 3 plus 10. 3 plus 10. Totally is what?
Totally is what? 13. 13. Final answer. Final answer. Check.
Check. Let's go to continue the example 5. Let's go back to the example 5. The same thing. The same thing.
I'm going to rewrite the original question function twice. I'm going to rewrite this original question function twice. And we can make the proper signs right on that.
And we can make the proper signs right on that. Equal to Equal to 3 minus. 3 minus 0.5 squared minus. 3 minus 0.5 x cubed, 0.4 squared, minus 2x, that's one time, That's one time plus the second time.
plus the second time, 3 minus 0.56 cubed 0.4 squared 2x. 3 minus 0.5. x cubed, 0.4, x squared, minus 2x. Now, we're going to find the derivative of each one. Now, we're going to find the derivative of each one.
The first one, The first one, you're going to find the derivative of this. you're going to find the derivative of this. Second time, Second time, you're going to find the derivative of this one.
you're going to find the derivative of this one. Let's solve it. Let's solve it.
Make sure you always bring down that equal sign. Make sure you always bring down the equal sign. Okay?
Okay? A lot of people, A lot of people, a lot of students are scared by the decimal point. a lot of students are scared by the decimal point. Okay? It's okay.
It's okay. We can do it too. We can do it too.
This one. This one. You have your calculator available on site too. You have your calculator available on site too.
First parenthesis, first parentheses you don't touch it just bring it down you don't touch it. Just burn it down. 3 minus 3 minus 0.5 x cubed times let's find the derivative of the 0.4 x squared so basically you are going to multiply 0.4 twice 0.4 twice treat 0.5x cubed times. Let's find the derivative of the 0.4x squared. So basically you are going to multiply 0.4 twice.
0.4 twice. Treat it as a 40 cents twice. it as a 40 cents twice that make what That makes what? 80 cents x to the power you have to make sure you're minus 1 on the exponent 80 cents, x to the power, you have to make sure you get a minus 1 from the exponent, that give me x to the 1. that give me x to the 1. minus Minus 2 times this x has exponent 1, 2 times this x has an exponent of 1. so 2 times 1, So 2 times 1, it's a 2. it's a 2. And you know that the exponent right here, And you know that the exponent right here, you are going to minus 1. you are going to minus 1. That will give me an x to the power of 0. That will give me an x to the power of 0. Repeat one more time. one more time any number to the power 0 is gonna give me a 1 so you just finish the derivative for the first part before the plus now we're plus find the derivative of the Any number to the power of 0 is going to give me a 1. So you just finished the derivative for the first part before the plus.
Now we're plus. Find the derivative of the second portion. the second portion I'm going to underline here for you. I'm going to underline here for you. Finding the derivative of the 3 constant, Finding the derivative of the 3 constant, it doesn't have the variable, it doesn't have the variable, it goes away, it goes away, minus.
minus, So you are going to multiply... so you are going to multiply 0.53 times, 0.5 three times which means you are multiplying the 50 cents three times 50 cents three times that means what buck and half which means you are multiplying the 50 cents, 3 times, 50 cents, 3 times, that means what? Back in half, x to the power, X to the power you are going to minus one from this you are going to minus 1 from the exponent, exponent that's a 2 times you need to bring down the point for x square minus 2 times, you need to bring down the 0.4x squared minus 2x you just finish all the derivative all you have to do the next step is to for your out and combine the like terms so let's do it we're going to pull you out and combine like terms it's 2x.
You just finished all the derivative. All you have to do, the next step, is to FOIL out and combine the like terms. So let's do it. We're going to FOIL out and combine like terms.
It's going to be equal to... gonna be equal to For your out, For your out, 3 times 0.8 is 3 times 0.8 is 2.4x. 2.4x.
3 times negative 2, 3 times negative 2, negative 6. negative 6, Minus. minus, And. and negative negative 0.5x times 0.8. 0.5x times 0.8.
You can read it as 0.5 as a You can read it as 0.5 as a 50%. Half of the 0.8. 50%. Half of the 0.8.
Basically, Basically, it's going to be half of the it's going to be half of the 80 cents is 40 cents. 80 cents is 40 cents. X to the power. X to the power, Don't forget, don't forget you need to add them. you need to add them.
3 plus 1 is a 4. 3 plus 1 is a 4. Last one. Last one. Negative times negative is a positive Negative times negative is a positive.
50 cents twice that made me a buck next to this there and we are going to repeat the second part once again plus you 50 cents. Twice. That made me a buck. Let's do that again. And we are going to repeat the second part once again.
Plus. You are going to distribute. are going to distribute i'm going to use a different color marker you can see better because negative times positive is going to be what negative I'm going to use a different color marker.
You can see better. See what's... Negative times positive is going to be what? Negative.
1.5 times 0.4 is equal to negative 0.6. 1.5 times 0.4 is equal to negative... Point six.
Don't forget to carry down the x. Don't forget to carry down the x. You add the exponent, You add the exponent.
2 plus 2 is 4. Two plus two is four. And then the last one here, And then the last one here. negative times negative is positive. Negative times negative is positive. but can have twice that will make me a three dollars x to the power make sure you add the exponent once again two plus one three parentheses and you just finish all the work the final final Bucking half twice, that will make me $3.
x to the power, make sure you add the exponent once again. 2 plus 1, 3, parentheses. And you just finished all the work. The final step is to combine the like terms right here. step is to combine the like terms right here you are going to start with the highest exponent which is x to the fourth You are going to start with the highest exponent, which is x to the fourth.
Negative 0.4 plus negative 0.6. Negative 0.4 plus negative. 0.6. Basically, Basically it's going to be a negative it's going to be a negative 1x to the 4th.
1x to the 4th. Next, Next, and combine the x cubes. and combine the x cubes.
Here comes Here comes 1x cubed plus 3 more. 1x cubed plus 3 more. So you have plus So you have plus 4x cubed. 4x cubed. And then nowhere else you can find anything to combine with the And then nowhere else you can find anything to combine with the 2.4x.
So you bring it down. 2.4x. So you burn it down. Plus Plus 2.4x.
2.4x. Final term to minus 6. Final term to minus 6. Final answer. Final answer. Check.
Check.