Stony Brook MAT 122 Fall 2017
Lecture 18: The Product and Quotient Rules
October 9, 2017

Start   Now as I said the first part of the course was really a lot of this sort of function algebra review, some limits, and a basic idea of derivatives. Now we're going to spend a lot of time on how to find the derivative of different things.
and then using them for problems.
Then we'll go back and do anti-derivatives which is harder than derivatives.
It's called integration. Integration is definitely harder than differentiation.
The good news is we don't spend as much time on it.
0:31So there's a midterm in about 3 weeks I think.
3 and a half weeks I think. So the second midterm is very close to the first midterm.
Which means we won't have as much material to cover.
And then we have Thanksgiving and then Christmas and then we're done.
Okay so it's not so bad.
Six weeks till Thanksgiving.
Lights, more lights.
Okay I think we're at full lights.
Okay so remember how to find the derivative of just something to a power.
1:02So the derivative of 5x^3 or something like that.
And all you do is bring the power in front, multiply, and reduce.
It's a very primitive function when you have something like 5x^3.
Not a lot is going on. So now we're going to start to work with more complicated types of problems.
So the first rule we're going to learn is called the product rule.
1:39The product rule. This is very useful when you have two functions that are multiplied by each other.
Okay? They're multiplying each other. However you say that grammatically.
Okay so you have function number one and function number two.
For example if you have y equals a function times another function so f(x) * g(x).
2:07The derivative is not just the derivative of this times the derivative of that.
That would be too easy.
And if you- we're not going to derive it but if you actually sort of look at the math it's harder to do than you think.
So there's a formula. It's not that complicated.
So the derivative is f(x) g'(x) + g(x) f'(x).
2:39Alright so it's the first function times the derivative of the second function plus the second function times the derivative of the first function.
Or the other way around.
The order doesn't really matter because you're multiplying and you're adding.
So a+b is the same as b+a.
You just have to do this so we'll do some examples now to make sure you can handle it.
3:03And the product rule will be very useful when you have functions with logarithms and exponentials. And you often end up with a lot of products.
The type we practice now, there's less of them. We'll see.
So say y is... Now you certainly could just multiply this out.
But we'll practice it with the product rule first.
3:34The derivative is- you leave the first function alone.
And you multiply it by the second function.
Plus, and now you go the other way.
You leave the second function alone, times the derivative of the first function.
4:06Now if I gave this to you as a homework problem or a test question you would not be expected to simplify this.
Just leave it alone okay?
And one of the good things about getting up to this level of math is we don't care if you simplify.
Or if you combine all of these things. However, just to show you that it works I'm going to multiply all of this out.
So you get 12x^5 +60x^4 +6x^5 +15x^4 -4x -10
4:43And that simplifies to 18x^5 +75x^4 -4x -10 Why would you want to actually do that in real life? Well you'd want to do that if you had to take the derivative again.
That's called the second derivative.
5:01Because this would not be fun to take the derivative of and this would be much easier to take the derivative of.
Otherwise let's say I wanted to find 0's you'd actually rather have it in this form.
This is easier for finding 0's.
Okay well graphing, some other stuff.
Okay so most of the time simplifying really doesn't make your life simpler.
That's why it's a bad phrase.
Alright so now let's show that this works.
Suppose instead we first multiplied it out then took the derivative.
5:34We should of course get the same answer.
So let's see.
We have 3x^6 -2x^2 +15x^5 -10x.
6:08So dy/dx= 18x^5 -4x +75x^4 -10.
That worked. Alright so let's give you guys a couple to practice.
Here's why you'd want to use the product rule.
6:45Because you wouldn't want to do that- you wouldn't want to multiply that out first.
So you want to be careful. Sometimes you do want to multiply it out, sometimes you don't.
7:12Alright why doesn't everybody take care of that.
See how you do on those.
9:50You got it?
10:02Ready? It's really not hard because remember we've done the derivatives of polynomials.
Take the first function, you do nothing.
Then we multiply it by the second function, which is 12x^3 + 33x +4.
That's the first function times the derivative of the second function.
10:32Yes 33x^2 sorry. Thank you.
And by the way, one of the things that all of you guys do and I do as well, it's very easy to mess up something small on these problems.
So I remind the TAs of that.
I hope they paid attention when they graded your papers. We'll see.
Plus the reverse.
(24x^2 +12) (3x^4 +11x^3 + 4x-3)
11:10And by the way you could do it the other order.
That was your question right. So you could've done this term and then this term.
The order doesn't matter with the product rule.
Sure you could also reverse those.
Because you're just multiplying. And with this kind of stuff as I said a*b is the same as b*a.
11:33And a+b is the same thing as b+a.
So if you said I have AB + CD you could also have CD + AB.
You could have BA+CD and so on and so forth. They're all the same thing.
With the quotient rule which is coming up in a minute the order is more important.
Alright the product rule again.
So this one, now here I might multiply that out first.
12:03Because you want to get some canceling and rearranging and all of that.
However if you're not brave enough to multiply that out and I don't blame you, then you're good.
So leave the first function alone.
Multiply by the second.
Remember we memorized the derivative of √x which is 1/2√x.
12:30Or at least I memorized it.
Ignore that. Times the derivative of this.
Plus, now the derivative okay?
So (x^3 + 1/x + x)(2x -1/2√x).
13:15I say it I don't always write it.
Okay how are we doing so far?
Good? You guys get your midterm grades by the way?
How'd you do?
I haven't run the numbers yet so I don't know what's a good grade and what's a bad grade.
13:31As I told you guys on Friday I don't establish the letter grades at this point.
Because that's not healthy. So you get your grade and then at the end of the course we figure out what's worth an A, what's worth a B.
Otherwise you might do worse than you're thinking or better than you're thinking.
Based on where things are now.
What I can do is give you an idea of how you did relative to the class as a whole.
I'll run that today or tomorrow and then I'll give you guys the statistics on Wednesday.
14:00Alright how do we feel about the product rule?
It's not so bad.
Then there's something else called the quotient rule.
So product is f*g, quotient is f/g.
It would be really nice if the quotient was just if you stuck a minus sign in there.
Because that seems to be what happens a lot. Unfortunately it's not that simple.
14:34So y= f(x)/g(x) And dy/dx is you take g(x)f'(x) - f(x)g'(x) So it looks just like the product rule with a minus sign, over the bottom function squared.
15:03When we do chain rule you'll understand why.
So with the quotient rule getting the order correct is very important.
Okay? It's the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function over the bottom function squared.
So there's a little pneumonic that some people teach which is LoDeHi - HiDeLo / Lo^2.
15:36So LoDeHi-HiDeLo so the lower function, derivative of the higher function minus the higher function, derivative of the lower function.
Okay? And as I said this one causes many people more problems.
And again I can derive it for you, show you where it comes from.
It requires playing with that definition of derivatives a lot.
16:08So if we had There's really no way to simplify that in advance.
Not that I can think of.
And there's really nothing that'll make it simpler.
16:30So the derivative, first you go to the bottom function.
Times the derivative of the top function.
And by the way here you could switch the order. This stuff doesn't matter.
However it's this one minus the other way. So the top function times the derivative of the bottom function.
So it's very important that you get this term minus that term not the other way around.
17:03You'll end up with a negative in the answer otherwise.
And with these you almost never want to simplify.
Because it's a lot of work to simplify. You would if you wanted to do another derivative because this is messy.
Okay? But other than that you leave it alone.
17:31Usually what happens with derivatives is 3 things.
You keep going so sometimes you're just asked to take the derivative.
Otherwise you either take another derivative, you plug in a value, or you find where the function is 0.
So finding where this is 0 requires setting the numerator equal to 0 because a fraction is 0 when the numerator is 0.
Not the denominator.
And that's messy. I wouldn't want to find where this is 0 but at least factor 4.
18:02The second thing is plugging in numbers well I might as well plug in a number here.
Why would you simplify?
First plug in the number and then simplify. It's much easier because then you're down to just arithmetic.
But if I had to take another derivative I would want to clean this up as much as possible first.
I certainly would multiply this out and this out and combine like terms.
Alright let's do another one of these.
18:37Sometimes the simplifying is not that tough.
Now suppose we had something like this.
You could do polynomial division first.
Remember that? You divide the bottom and the top and you get some terms, you get a leftover term, but why?
19:05I wouldn't bother.
So it's LoDeHi - HiDeLo so it's the bottom function minus the derivative of the top function. Oops, times the derivative of the top function.
19:33Minus the top function times the derivative of the bottom function.
Over the bottom function squared.
20:01Now here let's just show you what happens when you simplify. Sometimes simplifying isn't so bad.
Because you'll get lots of cancelation.
So if I had to find say when the numerator was 0 you look at that and you say well that's really unpleasant.
You have to multiply out the numerator. We're going to get (30x^3 -9x^2 +50x^2 -15x) - (30x^3 +25x^2 -18x^2 -15x).
20:51Which simplifies to- over the bottom stuff.
30x^3 +41x^2 -15x -30x^3 -7x^2 +15x.
21:16Lots of multiplying out. This is why you learn to factor in the 8th grade. 7th or 8th grade.
Factoring and combining- bleh.
Oh look the 30x^3's cancel.
21:32And the 15x's cancel so now the numerator is just 34x^2.
So the derivative ends up 34x^2 / (3x^2 +5x)^2.
So that's certainly easier right?
So if I had to find 0's well it's 0 at 0.
When x is 0 this thing comes out 0.
22:00Actually it's undefined because the bottom is also 0.
You could foil out the bottom, foil out an x^2 and get some more canceling.
But this is undefined at 0 so it's kind of a weird function. I just made it up.
You got it? You with me?
Good let's give you a couple to practice.
22:58Find the derivative of that one.
23:24And that one.
27:27Alright that's probably long enough.
27:37Okay so a couple of people asked me, by the way, one the exam score is out of 85.
Not out of 100. I don't believe that's a percentage that the TAs put up I think they just put up your raw score.
So if you got a 77 it's not 77% it's 77 out of 85.
I just can't tell you what that turns into.
Okay? Because I don't know and I'm not going to figure it out now.
Another question somebody asked me is will we be needed to simplify, would I asked you to simplify these.
28:05I don't expect to. Unless there's one where I think you get a really funny answer when you simplify, that's 0.
Um you should be able to, as I said, take another derivative, plug in 0, or find where it's 0.
So those are the 3 skills you're going to have to be able to do.
One of the problems people have with calculus in general is they can take a derivative.
It's not really very hard it's just a mechanical skill.
28:30But then they have problems with the rest of the problem.
They don't manipulate correctly, they can't solve it, they can't factor, rearrange and all those things.
So their algebra let's them down, not the calculus.
Okay so the derivative.
The bottom function times the derivative of the top function.
29:00Minus top function times the derivative of the bottom function.
Over bottom function squared.
Okay, so again.
Bottom times derivative of the top minus top times derivative of the bottom over the bottom squared.
29:36So far so good?
This one, by the way, this one you could simplify it first.
And if you were good at your algebra you basically divide the bottom into the top.
And you get a simpler expression then the derivative is actually pretty easy.
But we haven't done that so I'm not going to do that with you.
So let's just do the derivative.
30:05Bottom times the derivative of the top.
Minus the top function ties the derivative of the bottom function.
Over bottom function squared.
Now just for laughs how would you simplify that?
30:33Well you have two routes. One is distribute out the top.
And then cancel. The other is do some canceling first.
So as I said just enforce your algebra skills.
Factor 1 out of 2√x out of the top.
And you're left with (√x +2)-(√x -2)
31:05Over this squared.
You take the 1 out of 2√x out of both terms right?
And then you get √x +2 - √x -2 so √x isn't going to cancel.
So you get 1/2√x times- these cancel, 2- -2 is 4.
31:43So then this just moves into the denominator so this would be 4, well the 4 out of 2 cancel but we'll do that in a second.
2√x times the (√x +2)^2.
Or 2/√x (√x +2)^2
32:04As I said why would you want to do this? Well first of all if I said where are the zeros.
There are no zeros. Because the numerator is never zero.
The denominator is 0 so your problem at x=0 is it's undefined there.
There's no place where this is going to be zero.
And in fact you could multiply this out and distribute the √x.
32:31So if you wanted to take another derivative of this, it would be much easier to take the second derivative of this than the second derivative of that.
It's much easier to do the 0's of this than the 0's of that but say I said let's just do this at x=9.
Well you can plug 9 in here.
You don't need to plug 9 in there. You get the same answer but why do all that algebra when instead you just plug 9 in and do the arithmetic.
Alright.
33:03Two more problems.
34:48Okay there's two fun ones.
39:15Alright let's start going over these.
So when we say find the equation of the tangent line there's two things you have to do.
You have to find the y-coordinate and you have to find the slope.
So first let's just find the y-coordinate.
When I plug in 1 I get (1-4+1)(1+8) which is -18.
39:38You all get -18?
Okay now to find the slope.
The derivative is (x^2 -4x +1)(3x^2 +8).
40:01Minus (x^3 +8x)(2x -4).
Everybody get that?
Now you plug in 1 because you want to find the slope.
So you say m for slope is 1-4+1- did I not take the derivative?
40:30That should be plus. Thank you.
You guys gotta raise your hands.
Times (3+8) + (1+8)(2-4) which is -22-18 which is -40.
Everybody get -40?
Anybody not get -40? Everybody else is just copying.
41:07Okay so then the equation is y+18= -40(x-1).
So on an exam question you get some points for finding y, you get some points for finding the derivative, you get some points for plugging in, you get some points for the equation.
41:33Alright same thing here.
So first we plug in 4 we get 4+2 over 4-2 which is 3.
Then for the derivative So remember this is why you don't simplify.
42:00You don't simplify because when you plug in 1 this is much easier to simplify than first doing the algebra and then plugging in 1.
Okay? So you always want to plug in first.
So here we have the bottom times the derivative of the top.
42:30Minus the top times the derivative of the bottom.
Over the bottom squared.
43:04And now we plug in x=4.
So you get (4-2)(1+1/4) - (4+2)(1-1/2) Over (4-2)^2.
43:32Did I make a mistake?
Oh 2√x. Yeah I left that out.
Okay 2 times 5/4 is 5/2.
44:016 * 3/4 =9/2.
Over 4. Which is -4/2 /4 which is -1/2.
You could do the arithmetic more slowly.
I won't give you anything too messy because you guys aren't good without calculators.
I understand.. So the equation is y-3= -1/2 (x-4).
44:34Okay we'll practice a couple more of these on Wednesday. So everybody have a nice day, stay dry.