Uploaded by TheIntegralCALC on 12.02.2010

Transcript:

Hey everybody!

Product rule...

Get excited!

We're going to do

a couple problems.

The first one: f(x)

equals (2x plus 3)

times (3x minus 2).

And, we just finished doing

some chain rule problems;

product rule is another one of those rules.

You use it all the time

so it's not necessarily

anything valuable by itself.

You use it to figure out at the problems

but you need to know it

and I'm sure that they'll put it on a test and ask you

to find the derivative. So...

Okay... So, product rule.

All it is,

you have multiple terms in a problem,

at least two but it can be... can be more,

right now we have two,

one is (2x plus 3) and the other one is (3x minus 2),

and

it's how you find the derivative

with these terms. So,

the way that we're going to do that,

let's go ahead and call it f'(x),

is you're going to take the derivative of the first one,

keeping the second one the same,

and then, take the derivative of the second one,

keeping the first one the same.

So,

let's go ahead and take the derivative of

(2x plus 3).

So, the derivative of (2x plus 3)

is just 2,

and then, you're going to multiply

by (3x minus 2), we keep that the same,

and then, you add... you add them together.

So, we took the derivative of this,

kept this the same,

now we need to take the derivative of this

and keep this the same.

So, the derivative of (3x minus 2)

is 3 and then we multiply by

(2x plus 3), keeping that the same.

So, that's...

that's product rule at its most basic:

taking the derivative of one term,

keeping all of the others the same

and, adding them together as you go.

So you could do this with three terms as well and we will.

Two other things to know...

You can

simplify this if you want to,

6x minus 4 plus 6x plus 9,

and then, 6x plus 6x,

12x,

-4 plus 9

is plus 5.

So,

this I think it is...

is a cleaner answer than this one up here.

The other thing is

if you had to...

if you would be asked this problem on a test

and it says demonstrate product rule,

you're going to want to do it this way,

if it just says find the derivative

and during... maybe not so comfortable with product rule,

obviously, you could multiply this out

and then take the derivative.

So you could say

2x times 3x,

6x^2,

let's see... minus 4x plus 9x

minus 6, which would get you

6x^2

(minus 4x plus 9x)... plus 5x minus 6.

So you could

multiply this out

and then take the derivative of that

if it's more comfortable for you,

but if you're asked to show product rule,

you're going to need to do it this way

and then

probably your choice about whether or not you want to simplify it to this point.

So...

that's the first one.

Product rule...

Get excited!

We're going to do

a couple problems.

The first one: f(x)

equals (2x plus 3)

times (3x minus 2).

And, we just finished doing

some chain rule problems;

product rule is another one of those rules.

You use it all the time

so it's not necessarily

anything valuable by itself.

You use it to figure out at the problems

but you need to know it

and I'm sure that they'll put it on a test and ask you

to find the derivative. So...

Okay... So, product rule.

All it is,

you have multiple terms in a problem,

at least two but it can be... can be more,

right now we have two,

one is (2x plus 3) and the other one is (3x minus 2),

and

it's how you find the derivative

with these terms. So,

the way that we're going to do that,

let's go ahead and call it f'(x),

is you're going to take the derivative of the first one,

keeping the second one the same,

and then, take the derivative of the second one,

keeping the first one the same.

So,

let's go ahead and take the derivative of

(2x plus 3).

So, the derivative of (2x plus 3)

is just 2,

and then, you're going to multiply

by (3x minus 2), we keep that the same,

and then, you add... you add them together.

So, we took the derivative of this,

kept this the same,

now we need to take the derivative of this

and keep this the same.

So, the derivative of (3x minus 2)

is 3 and then we multiply by

(2x plus 3), keeping that the same.

So, that's...

that's product rule at its most basic:

taking the derivative of one term,

keeping all of the others the same

and, adding them together as you go.

So you could do this with three terms as well and we will.

Two other things to know...

You can

simplify this if you want to,

6x minus 4 plus 6x plus 9,

and then, 6x plus 6x,

12x,

-4 plus 9

is plus 5.

So,

this I think it is...

is a cleaner answer than this one up here.

The other thing is

if you had to...

if you would be asked this problem on a test

and it says demonstrate product rule,

you're going to want to do it this way,

if it just says find the derivative

and during... maybe not so comfortable with product rule,

obviously, you could multiply this out

and then take the derivative.

So you could say

2x times 3x,

6x^2,

let's see... minus 4x plus 9x

minus 6, which would get you

6x^2

(minus 4x plus 9x)... plus 5x minus 6.

So you could

multiply this out

and then take the derivative of that

if it's more comfortable for you,

but if you're asked to show product rule,

you're going to need to do it this way

and then

probably your choice about whether or not you want to simplify it to this point.

So...

that's the first one.