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.
