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PROFESSOR: Hi.

2
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Well, today's the chain rule.

3
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Very, very useful rule, and it's
kind of neat, natural.

4
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Can I explain what a chain
of functions is?

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There is a chain of functions.

6
00:00:21,810 --> 00:00:26,360
And then we want to know the
slope, the derivative.

7
00:00:26,360 --> 00:00:28,600
So how does the chain work?

8
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So there x is the input.

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It goes into a function
g of x.

10
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We could call that inside
function y.

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So the first step
is y is g of x.

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So we get an output
from g, call it y.

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That's halfway, because that y
then becomes the input to f.

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That completes the chain.

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It starts with x, produces
y, which is the inside

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function g of x.

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And then let me call
it z is f of y.

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And what I want to know
is how quickly does

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z change as x changes?

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That's what the chain
rule asks.

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It's the slope of that chain.

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Can I maybe just tell
you the chain rule?

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And then we'll try it
on some examples.

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You'll see how it works.

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OK, here it is.

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The derivative, the slope of
this chain dz dx, notice I

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want the change in the whole
thing when I change the

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original input.

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Then the formula is that
I take-- it's nice.

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You take dz dy times dy dx.

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So the derivative that we're
looking for, the slope, the

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speed, is a product of two
simpler derivatives that we

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probably know.

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And when we put the chain
together, we multiply those

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derivatives.

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But there's one catch
that I'll explain.

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I can give you a hint
right away.

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dz dy, this first factor,
depends on y.

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But we're looking for the change
due to the original

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change in x.

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When I find dz dy, I'm going
to have to get back to x.

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Let me just do an example
with a picture.

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You'll see why I
have to do it.

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So let the chain be cosine
of-- oh, sine.

45
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Why not?

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Sine of 3x.

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Let me take sine of 3x.

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So that's my sine of 3x.

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I would like to know if that's
my function, and I can draw it

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and will draw it, what
is the slope?

51
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OK, so what's the
inside function?

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What's y here?

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Well, it's sitting there
in parentheses.

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Often it's in parentheses so
we identify it right away.

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y is 3x.

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That's the inside function.

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And then the outside function
is the sine of y.

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So what's the derivative
by the chain rule?

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I'm ready to use the chain rule,
because these are such

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simple functions, I know their
separate derivative.

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So if this whole thing
is z, the chain rule

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says that dz dx is--

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I'm using this rule.

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I first name dz dy, the
derivative of z with respect

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to y, which is cosine of y.

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And then the second factor is
dy dx, and that's just a

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straight line with slope
3, so dy dx is 3.

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Good.

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Good, but not finished, because
I'm getting an answer

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that's still in terms of y, and
I have to get back to x,

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and no problem to do it.

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I know the link from y to x.

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So here's the 3.

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I can usually write it out here,
and then I wouldn't need

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parentheses.

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That's just that 3.

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Now the part I'm caring
about: cosine of 3x.

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Not cosine x, even though
this was just sine.

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But it was sine of y, and
therefore, we need cosine 3x.

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Let me draw a picture of this
function, and you'll see

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what's going on.

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If I draw a picture of--

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I'll start with a picture of
sine x, maybe out to 180

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degrees pi.

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This direction is now x.

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And this direction is going to
be-- well, there is the sine

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of x, but that's not
my function.

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My function is sine of 3x, and
it's worth realizing what's

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the difference.

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How does the graph change if
I have 3x instead of x?

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Well, things come sooner.

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Things are speeded up.

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Here at x equal pi, 180 degrees,
is when the sine

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goes back to 0.

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But for 3x, it'll be back to
0 already when x is 60

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degrees, pi over 3.

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So 1/3 of the way along, right
there, my sine 3x is this one.

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It's just like the sine
curve but faster.

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That was pi over 3 there,
60 degrees.

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So this is my z of x curve, and
you can see that the slope

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is steeper at the beginning.

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You can see that the slope--

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things are happening
three times faster.

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Things are compressed by 3.

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This sine curve is
compressed by 3.

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That makes it speed up so the
slope is 3 at the start, and I

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claim that it's 3
cosine of 3x.

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Oh, let's draw the slope.

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All right, draw the slope.

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All right, let me start with
the slope of sine-- so this

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was just old sine x.

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So its slope is just
cosine x along to--

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right?

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That's the slope starts at 1.

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This is now cosine x.

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But that's going out
to pi again.

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That's the slope of the original
one, not the slope of

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our function, of our chain.

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So the slope of our
chain will be--

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I mean, it doesn't
go out so far.

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It's all between here and
pi over 3, right?

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Our function, the one
we're looking at, is

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just on this part.

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And the slope starts out at 3,
and it's three times bigger,

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so it's going to be--
well, I'll just

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about get it on there.

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It's going to go down.

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I don't know if that's great,
but it maybe makes the point

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that I started up here at 3, and
I ended down here at minus

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3 when x was 60 degrees
because then--

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you see, this is a picture
of 3 cosine of 3x.

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I had to replace y by
3x at this point.

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OK, let me do two or three
more examples,

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just so you see it.

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Let's take an easy one.

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Suppose z is x cubed squared.

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All right, here is the
inside function.

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y is x cubed, and z is--
do you see what z is?

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z is x cubed squared.

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So x cubed is the
inside function.

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What's the outside function?

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It's a function of y.

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I'm not going to write--

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it's going to be the
squaring function.

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That's what we do outside.

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I'm not going to write
x squared.

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It's y squared.

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This is y.

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It's y squared that
gets squared.

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Then the derivative dz dx by
the chain rule is dz dy.

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Shall I remember
the chain rule?

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dz dy, dy dx.

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Easy to remember because in the
mind of everybody, these

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dy's, you see that they're
sort of canceling.

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So what's dz dy?

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z is y squared, so this
is 2y, that factor.

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What's dy dx?

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y is x cubed.

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We know the derivative
of x cubed.

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It's 3x squared.

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There is the answer, but it's
not final because I've got a y

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here that doesn't belong.

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I've got to get it back to.

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X So I have all together 2 times
3 is making 6, and that

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y, I have to go back and see
what was y in terms of x.

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It was x cubed.

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So I have x cubed there, and
here's an x squared,

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altogether x to the fifth.

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Now, is that the right answer?

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In this example, we can
certainly check it because we

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know what x cubed squared is.

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So x cubed is x times x times
x, and I'm squaring that.

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I'm multiplying by itself.

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There's another x
times x times x.

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Altogether I have x to
the sixth power.

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Notice I don't add those.

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When I'm squaring x cubed,
I multiply the 2 by

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the 3 and get 6.

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So z is x to the sixth, and of
course, the derivative of x to

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the sixth is 6 times x to the
fifth, one power lower.

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OK, I want to do two
more examples.

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Let me do one more right away
while we're on a roll.

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I'll bring down that board and
take this function, just so

184
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you can spot the
inside function

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and the outside function.

186
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So my function z is going to be
1 over the square root of 1

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minus x squared.

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Such things come up pretty often
so we have to know its

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derivative.

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We could graph it.

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That's a perfectly reasonable
function, and

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it's a perfect chain.

193
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The first point is to identify
what's the inside function and

194
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what's the outside.

195
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So inside I'm seeing this
1 minus x squared.

196
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That's the quantity that it'll
be much simpler if I just give

197
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that a single name y.

198
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And then what's the
outside function?

199
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What am I doing to this y?

200
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I'm taking its square root,
so I have y to the 1/2.

201
00:12:26,480 --> 00:12:29,870
But that square root is
in the denominator.

202
00:12:29,870 --> 00:12:33,520
I'm dividing, so it's
y to the minus 1/2.

203
00:12:33,520 --> 00:12:38,800
So z is y to the minus 1/2.

204
00:12:38,800 --> 00:12:42,150
OK, those are functions I'm
totally happy with.

205
00:12:42,150 --> 00:12:44,355
The derivative is what?

206
00:12:48,310 --> 00:12:51,210
dz dy, I won't repeat
the chain rule.

207
00:12:51,210 --> 00:12:53,090
You've got that clearly
in mind.

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It's right above.

209
00:12:54,490 --> 00:12:56,970
Let's just put in
the answer here.

210
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dz dy, the derivative, that's
y to some power, so I get

211
00:13:01,140 --> 00:13:05,306
minus 1/2 times y
to what power?

212
00:13:08,210 --> 00:13:10,560
I always go one power lower.

213
00:13:10,560 --> 00:13:12,355
Here the power is minus 1/2.

214
00:13:12,355 --> 00:13:17,490
If I go down by one, I'll
have minus 3/2.

215
00:13:17,490 --> 00:13:22,630
And then I have to have
dy dx, which is easy.

216
00:13:22,630 --> 00:13:25,430
dy dx, y is 1 minus x squared.

217
00:13:25,430 --> 00:13:31,030
The derivative of that
is just minus 2x.

218
00:13:31,030 --> 00:13:37,490
And now I have to assemble
these, put them together, and

219
00:13:37,490 --> 00:13:39,660
get rid of the y.

220
00:13:39,660 --> 00:13:42,490
So the minus 2 cancels
the minus 1/2.

221
00:13:42,490 --> 00:13:43,600
That's nice.

222
00:13:43,600 --> 00:13:50,560
I have an x still here, and
I have y to the minus 3/2.

223
00:13:50,560 --> 00:13:52,310
What's that?

224
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I know what y is, 1 minus x
squared, and so it's that to

225
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the minus 3/2.

226
00:14:01,220 --> 00:14:02,760
I could write it that way.

227
00:14:02,760 --> 00:14:05,960
x times 1 minus x squared--

228
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that's the y--

229
00:14:07,410 --> 00:14:11,370
to the power minus 3/2.

230
00:14:11,370 --> 00:14:13,470
Maybe you like it that way.

231
00:14:13,470 --> 00:14:15,870
I'm totally OK with that.

232
00:14:15,870 --> 00:14:19,250
Or maybe you want
to see it as--

233
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this minus exponent
down here as 3/2.

234
00:14:28,310 --> 00:14:30,440
Either way, both good.

235
00:14:30,440 --> 00:14:34,650
OK, so that's one
more practice.

236
00:14:34,650 --> 00:14:37,970
and I've got one more in mind.

237
00:14:37,970 --> 00:14:42,310
But let me return to this board,
the starting board,

238
00:14:42,310 --> 00:14:50,060
just to justify where did this
chain rule come from.

239
00:14:50,060 --> 00:14:53,160
OK, where do derivatives
come from?

240
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Derivative always start with
small finite steps, with delta

241
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rather than d.

242
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So I start here, I make a change
in x, and I want to

243
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know the change in z.

244
00:15:13,660 --> 00:15:23,320
These are small, but not
zero, not darn small.

245
00:15:23,320 --> 00:15:30,860
OK, all right, those are true
quantities, and for those, I'm

246
00:15:30,860 --> 00:15:41,870
perfectly entitled to divide and
multiply by the change in

247
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y because there will
be a change in y.

248
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When I change x, that produces
a change in g of x.

249
00:15:49,070 --> 00:15:50,840
You remember this was the y.

250
00:15:54,000 --> 00:15:56,760
So this factor--

251
00:15:56,760 --> 00:16:00,310
well, first of all, that's
simply a true

252
00:16:00,310 --> 00:16:02,840
statement for fractions.

253
00:16:05,720 --> 00:16:07,080
But it's the right way.

254
00:16:07,080 --> 00:16:09,070
It's the way we want it.

255
00:16:09,070 --> 00:16:14,190
Because now when I show it, and
in words, it says when I

256
00:16:14,190 --> 00:16:19,130
change x a little, that produces
a change in y, and

257
00:16:19,130 --> 00:16:22,520
the change in y produces
a change in z.

258
00:16:22,520 --> 00:16:26,920
And it's the ratio that we're
after, the ratio between the

259
00:16:26,920 --> 00:16:29,190
original change and
the final change.

260
00:16:29,190 --> 00:16:32,130
So I just put the
inside change up

261
00:16:32,130 --> 00:16:34,020
and divide and multiply.

262
00:16:34,020 --> 00:16:37,490
OK, what am I going to do?

263
00:16:37,490 --> 00:16:41,940
What I always do, whatever body
does with derivatives at

264
00:16:41,940 --> 00:16:44,740
an instant, at a point.

265
00:16:44,740 --> 00:16:47,510
Let delta x go to 0.

266
00:16:47,510 --> 00:16:51,270
Now as delta x goes to 0, delta
y will go to 0, delta z

267
00:16:51,270 --> 00:16:55,650
will go to 0, and we get
a lot of zeroes over 0.

268
00:16:55,650 --> 00:16:59,870
That's what calculus is
prepared to live with.

269
00:16:59,870 --> 00:17:03,040
Because it keeps this ratio.

270
00:17:05,780 --> 00:17:13,050
It doesn't separately think
about 0 and then later 0.

271
00:17:13,050 --> 00:17:17,230
It's looking at the ratio
as things happen.

272
00:17:17,230 --> 00:17:19,619
And that ratio does
approach that.

273
00:17:19,619 --> 00:17:22,119
That was the definition
of the derivative.

274
00:17:22,119 --> 00:17:26,460
This ratio approaches that,
and we get the answer.

275
00:17:26,460 --> 00:17:31,490
This ratio approaches the
derivative we're after.

276
00:17:31,490 --> 00:17:34,370
That in a nutshell
is the thinking

277
00:17:34,370 --> 00:17:36,470
behind the chain rule.

278
00:17:36,470 --> 00:17:41,390
OK, I could discuss it further,
but that's the

279
00:17:41,390 --> 00:17:42,430
essence of it.

280
00:17:42,430 --> 00:17:51,730
OK, now I'm ready to do one more
example that isn't just

281
00:17:51,730 --> 00:17:52,470
so made up.

282
00:17:52,470 --> 00:17:55,750
It's an important one.

283
00:17:55,750 --> 00:18:00,090
And it's one I haven't
tackled before.

284
00:18:00,090 --> 00:18:10,960
My function is going to be e to
the minus x squared over 2.

285
00:18:10,960 --> 00:18:13,670
That's my function.

286
00:18:13,670 --> 00:18:16,670
Shall I call it z?

287
00:18:16,670 --> 00:18:19,080
That's my function of x.

288
00:18:19,080 --> 00:18:23,790
So I want you to identify the
inside function and the

289
00:18:23,790 --> 00:18:27,850
outside function in that change,
take the derivative,

290
00:18:27,850 --> 00:18:30,290
and then let's look at the
graph for this one.

291
00:18:30,290 --> 00:18:34,430
The graph of this one is a
familiar important graph.

292
00:18:34,430 --> 00:18:37,770
But it's quite an interesting
function.

293
00:18:37,770 --> 00:18:41,140
OK, so what are you
going to take?

294
00:18:41,140 --> 00:18:42,560
This often happens.

295
00:18:42,560 --> 00:18:47,550
We have e to the something,
e to some function.

296
00:18:47,550 --> 00:18:50,760
So that's our inside
function up there.

297
00:18:50,760 --> 00:18:56,390
Our function y, inside function,
is going to be minus

298
00:18:56,390 --> 00:19:01,310
x squared over 2, that quantity

299
00:19:01,310 --> 00:19:03,060
that's sitting up there.

300
00:19:03,060 --> 00:19:06,980
And then z, the outside
function, is

301
00:19:06,980 --> 00:19:11,550
just e to the y, right?

302
00:19:11,550 --> 00:19:15,650
So two very, very simple
functions have gone into this

303
00:19:15,650 --> 00:19:18,900
chain and produced this
e to the minus x

304
00:19:18,900 --> 00:19:20,830
squared over 2 function.

305
00:19:20,830 --> 00:19:24,940
OK, I'm going to ask you for
the derivative, and you're

306
00:19:24,940 --> 00:19:25,760
going to do it.

307
00:19:25,760 --> 00:19:27,500
No problem.

308
00:19:27,500 --> 00:19:32,430
So dz dx, let's use
the chain rule.

309
00:19:32,430 --> 00:19:36,420
Again, it's sitting
right above.

310
00:19:36,420 --> 00:19:41,450
dz dy, so I'm going to take the
slope, the derivative of

311
00:19:41,450 --> 00:19:46,550
the outside function dz dy,
which is e to the y.

312
00:19:46,550 --> 00:19:51,380
And that has that remarkable
property, which is why we care

313
00:19:51,380 --> 00:19:55,230
about it, why we named it,
why we created it.

314
00:19:55,230 --> 00:19:56,940
The derivative of
that is itself.

315
00:19:59,940 --> 00:20:03,670
And the derivative of minus
x squared over 2 is--

316
00:20:03,670 --> 00:20:05,920
that's a picnic, right?--

317
00:20:05,920 --> 00:20:07,510
is a minus.

318
00:20:07,510 --> 00:20:09,580
x squared, we'll
bring down a 2.

319
00:20:09,580 --> 00:20:12,640
Cancel that 2, it'll
be minus x.

320
00:20:12,640 --> 00:20:15,720
That's the derivative of
minus x squared over 2.

321
00:20:15,720 --> 00:20:18,240
Notice the result is negative.

322
00:20:18,240 --> 00:20:21,390
This function is at
least out where--

323
00:20:21,390 --> 00:20:25,460
if x is positive, the whole
slope is negative, and the

324
00:20:25,460 --> 00:20:27,650
graph is going downwards.

325
00:20:27,650 --> 00:20:29,290
And now what's--

326
00:20:29,290 --> 00:20:31,230
everybody knows this
final step.

327
00:20:31,230 --> 00:20:33,360
I can't leave the answer
like that because

328
00:20:33,360 --> 00:20:34,870
it's got a y in it.

329
00:20:34,870 --> 00:20:39,890
I have to put in what
y is, and it is--

330
00:20:39,890 --> 00:20:42,590
can I write the minus x first?

331
00:20:42,590 --> 00:20:47,090
Because it's easier to write it
in front of this e to the

332
00:20:47,090 --> 00:20:51,640
y, which is e to the minus
x squared over 2.

333
00:20:51,640 --> 00:20:54,360
So that's the derivative
we wanted.

334
00:20:54,360 --> 00:20:58,000
Now I want to think about
that function a bit.

335
00:20:58,000 --> 00:21:05,570
OK, notice that we started
with an e to the minus

336
00:21:05,570 --> 00:21:10,520
something, and we ended with
an e to the minus something

337
00:21:10,520 --> 00:21:12,320
with other factors.

338
00:21:12,320 --> 00:21:15,200
This is typical for
exponentials.

339
00:21:15,200 --> 00:21:20,080
Exponentials, the derivative
stays with that exponent.

340
00:21:20,080 --> 00:21:23,800
We could even take the
derivative of that, and we

341
00:21:23,800 --> 00:21:26,660
would again have some
expression.

342
00:21:26,660 --> 00:21:30,470
Well, let's do it in a minute,
the derivative of that.

343
00:21:30,470 --> 00:21:36,230
OK, I'd like to graph these
functions, the original

344
00:21:36,230 --> 00:21:42,690
function z and the slope
of the z function.

345
00:21:42,690 --> 00:21:45,910
OK, so let's see.

346
00:21:45,910 --> 00:21:48,000
x can have any sign.

347
00:21:50,780 --> 00:21:53,430
x can go for this--

348
00:21:53,430 --> 00:21:54,760
now, I'm graphing this.

349
00:21:58,590 --> 00:22:00,420
OK, so what do I expect?

350
00:22:00,420 --> 00:22:05,220
I can certainly figure out
the point x equals 0.

351
00:22:05,220 --> 00:22:12,030
At x equals 0, I have e
to the 0, which is 1.

352
00:22:12,030 --> 00:22:16,170
So at x equals 0, it's 1.

353
00:22:16,170 --> 00:22:20,830
OK, now at x equals to 1, it
has dropped to something.

354
00:22:23,660 --> 00:22:28,520
And also at x equals minus
1, notice the symmetry.

355
00:22:28,520 --> 00:22:32,800
This function is going to be--
this graph is going to be

356
00:22:32,800 --> 00:22:39,650
symmetric around the y-axis
because I've got x squared.

357
00:22:39,650 --> 00:22:42,010
The right official name
for that is we

358
00:22:42,010 --> 00:22:43,660
have an even function.

359
00:22:43,660 --> 00:22:48,950
It's even when it's same
for x and for minus x.

360
00:22:48,950 --> 00:22:51,830
OK, so what's happening
at x equal 1?

361
00:22:51,830 --> 00:22:54,920
That's e to the minus 1/2.

362
00:22:54,920 --> 00:22:55,690
Whew!

363
00:22:55,690 --> 00:22:57,720
I should have looked
ahead to figure out

364
00:22:57,720 --> 00:23:00,460
what that number is.

365
00:23:00,460 --> 00:23:01,540
Whatever.

366
00:23:01,540 --> 00:23:05,620
It's smaller than 1, certainly,
because it's e to

367
00:23:05,620 --> 00:23:07,210
the minus something.

368
00:23:07,210 --> 00:23:11,890
So let me put it there, and
it'll be here, too.

369
00:23:11,890 --> 00:23:16,740
And now rather than a particular
value, what's your

370
00:23:16,740 --> 00:23:18,240
impression of the whole graph?

371
00:23:23,060 --> 00:23:24,590
The whole graph is--It's
symmetric, so it's going to

372
00:23:24,590 --> 00:23:28,230
start like this, and it's
going to start sinking.

373
00:23:28,230 --> 00:23:30,200
And then it's going to sink.

374
00:23:30,200 --> 00:23:31,950
Let me try to get through
that point.

375
00:23:35,570 --> 00:23:36,870
Look here.

376
00:23:36,870 --> 00:23:43,150
As x gets large, say x is even
just 3 or 4 or 1000, I'm

377
00:23:43,150 --> 00:23:47,170
squaring it, so I'm getting
9 or 16 or 1000000.

378
00:23:47,170 --> 00:23:48,890
And then divide by 2.

379
00:23:48,890 --> 00:23:49,700
No problem.

380
00:23:49,700 --> 00:23:53,600
And then e to the minus is--

381
00:23:53,600 --> 00:23:57,590
I mean, so e to the thousandth
would be off

382
00:23:57,590 --> 00:24:00,780
that board by miles.

383
00:24:00,780 --> 00:24:06,560
e to the minus 1000 is a very
small number and getting

384
00:24:06,560 --> 00:24:11,560
smaller fast. So this is going
to get-- but never touches 0,

385
00:24:11,560 --> 00:24:13,240
so it's going to--

386
00:24:13,240 --> 00:24:17,550
well, let's see.

387
00:24:17,550 --> 00:24:22,490
I want to make it symmetric, and
then I want to somehow I

388
00:24:22,490 --> 00:24:27,870
made it touch because this
darn finite chalk.

389
00:24:27,870 --> 00:24:31,750
I couldn't leave
a little space.

390
00:24:31,750 --> 00:24:35,320
But to your eye it touches.

391
00:24:35,320 --> 00:24:41,100
If we had even fine print, you
couldn't see that distance.

392
00:24:41,100 --> 00:24:50,370
So this is that curve, which was
meant to be symmetric, is

393
00:24:50,370 --> 00:24:55,610
the famous bell-shaped curve.

394
00:24:58,320 --> 00:25:07,160
It's the most important curve
for gamblers, for

395
00:25:07,160 --> 00:25:10,740
mathematicians who work
in probability.

396
00:25:10,740 --> 00:25:15,000
That bell-shaped curve will come
up, and you'll see in a

397
00:25:15,000 --> 00:25:20,150
later lecture a connection
between how calculus enters in

398
00:25:20,150 --> 00:25:23,110
probability, and it enters
for this function.

399
00:25:23,110 --> 00:25:25,790
OK, now what's this slope?

400
00:25:25,790 --> 00:25:28,170
What's the slope of
that function?

401
00:25:28,170 --> 00:25:32,650
Again, symmetric, or maybe
anti-symmetric, because I have

402
00:25:32,650 --> 00:25:33,960
this factor x.

403
00:25:33,960 --> 00:25:35,290
So what's the slope?

404
00:25:35,290 --> 00:25:37,660
The slope starts at 0.

405
00:25:37,660 --> 00:25:40,000
So here's x again.

406
00:25:40,000 --> 00:25:45,270
I'm graphing now the slope,
so this was z.

407
00:25:45,270 --> 00:25:48,660
Now I'm going to graph
the slope of this.

408
00:25:48,660 --> 00:25:55,200
OK, the slope starts out at 0,
as we see from this picture.

409
00:25:55,200 --> 00:25:59,570
Now we can see, as I go forward
here, the slope is

410
00:25:59,570 --> 00:26:01,540
always negative.

411
00:26:01,540 --> 00:26:03,480
The slope is going down.

412
00:26:03,480 --> 00:26:09,380
Here it starts out--

413
00:26:09,380 --> 00:26:13,710
yeah, so the slope is 0 there.

414
00:26:13,710 --> 00:26:16,085
The slope is becoming more
and more negative.

415
00:26:19,545 --> 00:26:20,400
Let's see.

416
00:26:20,400 --> 00:26:23,260
The slope is becoming more and
more negative, maybe up to

417
00:26:23,260 --> 00:26:25,660
some point.

418
00:26:25,660 --> 00:26:29,460
Actually, I believe it's
that point where

419
00:26:29,460 --> 00:26:31,220
the slope is becoming--

420
00:26:31,220 --> 00:26:32,990
then it becomes less negative.

421
00:26:32,990 --> 00:26:33,990
It's always negative.

422
00:26:33,990 --> 00:26:43,030
I think that the slope goes down
to that point x equals 1,

423
00:26:43,030 --> 00:26:46,570
and that's where the slope
is as steep as it gets.

424
00:26:46,570 --> 00:26:51,110
And then the slope comes
up again, but the slope

425
00:26:51,110 --> 00:26:53,740
never gets to 0.

426
00:26:53,740 --> 00:26:57,430
We're always going downhill,
but very slightly.

427
00:26:57,430 --> 00:27:02,170
Oh, well, of course, I expect
to be close to that line

428
00:27:02,170 --> 00:27:04,600
because this e to the minus
x squared over 2

429
00:27:04,600 --> 00:27:06,930
is getting so small.

430
00:27:06,930 --> 00:27:12,470
And then over here, I think
this will be symmetric.

431
00:27:12,470 --> 00:27:15,040
Here the slopes are positive.

432
00:27:15,040 --> 00:27:16,210
Ah!

433
00:27:16,210 --> 00:27:17,980
Look at that!

434
00:27:17,980 --> 00:27:22,890
Here we had an even function,
symmetric across 0.

435
00:27:22,890 --> 00:27:26,350
Here its slope turns
out to be--

436
00:27:26,350 --> 00:27:28,550
and this could not
be an accident.

437
00:27:28,550 --> 00:27:32,470
Its slope turns out to
be an odd function,

438
00:27:32,470 --> 00:27:34,660
anti-symmetric across 0.

439
00:27:34,660 --> 00:27:36,080
Now, it just was.

440
00:27:36,080 --> 00:27:39,480
This is an odd function, because
if I change x, I

441
00:27:39,480 --> 00:27:41,740
change the sign of
that function.

442
00:27:41,740 --> 00:27:48,140
OK, now if you will give me
another moment, I'll ask you

443
00:27:48,140 --> 00:27:49,910
about the second derivative.

444
00:27:49,910 --> 00:27:52,180
Maybe this is the first time
we've done the second

445
00:27:52,180 --> 00:27:54,040
derivative.

446
00:27:54,040 --> 00:27:56,760
What do you think the second
derivative is?

447
00:27:56,760 --> 00:28:00,990
The second derivative is the
derivative of the derivative,

448
00:28:00,990 --> 00:28:04,440
the slope of the slope.

449
00:28:04,440 --> 00:28:09,210
My classical calculus problem
starts with function one,

450
00:28:09,210 --> 00:28:14,130
produces function two,
height to slope.

451
00:28:14,130 --> 00:28:17,910
Now when I take another
derivative, I'm starting with

452
00:28:17,910 --> 00:28:23,440
this function one, and over here
will be a function two.

453
00:28:23,440 --> 00:28:31,910
So this was dz dx, and now here
is going to be the second

454
00:28:31,910 --> 00:28:32,050
derivative.

455
00:28:32,050 --> 00:28:33,300
Second derivative.

456
00:28:35,680 --> 00:28:42,940
And we'll give it a nice
notation, nice symbol.

457
00:28:42,940 --> 00:28:47,040
It's not dz dx, all squared.

458
00:28:47,040 --> 00:28:48,880
That's not what I'm doing.

459
00:28:48,880 --> 00:28:52,530
I'm taking the derivative
of this.

460
00:28:52,530 --> 00:28:54,240
So I'm taking--

461
00:28:54,240 --> 00:29:00,050
well, the derivative
of that, I could--

462
00:29:00,050 --> 00:29:05,090
I'm going to give a whole
return to the second

463
00:29:05,090 --> 00:29:05,530
derivative.

464
00:29:05,530 --> 00:29:07,700
It's a big deal.

465
00:29:07,700 --> 00:29:14,250
I'll just say how I write
it: dz dx squared.

466
00:29:14,250 --> 00:29:15,780
That's the second derivative.

467
00:29:15,780 --> 00:29:18,760
It's the slope of
this function.

468
00:29:18,760 --> 00:29:23,110
And I guess what I want is would
you know how to take the

469
00:29:23,110 --> 00:29:26,360
slope of that function?

470
00:29:26,360 --> 00:29:29,000
Can we just think what
would go into that,

471
00:29:29,000 --> 00:29:30,540
and I'll put it here?

472
00:29:30,540 --> 00:29:32,430
Let me put that function here.

473
00:29:32,430 --> 00:29:37,500
minus x e to the minus
x squared over 2.

474
00:29:37,500 --> 00:29:40,750
Slope of that, derivative
of that.

475
00:29:40,750 --> 00:29:42,990
What do I see there?

476
00:29:42,990 --> 00:29:44,500
I see a product.

477
00:29:44,500 --> 00:29:47,760
I see that times that.

478
00:29:47,760 --> 00:29:50,580
So I'm going to use
the product rule.

479
00:29:50,580 --> 00:29:55,280
But then I also see that in this
factor, in this minus x

480
00:29:55,280 --> 00:29:58,330
squared over 2, I see a chain.

481
00:29:58,330 --> 00:30:01,530
In fact, it's exactly
my original chain.

482
00:30:01,530 --> 00:30:03,790
I know how to deal
with that chain.

483
00:30:03,790 --> 00:30:05,710
So I'm going to use
the product rule

484
00:30:05,710 --> 00:30:08,060
and the chain rule.

485
00:30:08,060 --> 00:30:14,430
And that's the point that once
we have our list of rules,

486
00:30:14,430 --> 00:30:19,800
these are now what we might
call four simple rules.

487
00:30:19,800 --> 00:30:25,270
We know those guys: sum,
difference, product, quotient.

488
00:30:25,270 --> 00:30:28,390
And now we're doing the chain
rule, but we have to be

489
00:30:28,390 --> 00:30:33,070
prepared as here for a product,
and then one of these

490
00:30:33,070 --> 00:30:34,400
factors is a chain.

491
00:30:34,400 --> 00:30:36,510
All right, can we do it?

492
00:30:36,510 --> 00:30:38,185
So the derivative, slope.

493
00:30:41,300 --> 00:30:46,890
Well, slope of slope, because
this was the original slope.

494
00:30:46,890 --> 00:30:52,580
OK, so it's the first factor
times the derivative of the

495
00:30:52,580 --> 00:30:55,100
second factor.

496
00:30:55,100 --> 00:30:57,990
And that's the chain,
but that's the one

497
00:30:57,990 --> 00:30:59,410
we've already done.

498
00:30:59,410 --> 00:31:03,530
So the derivative of that is
what we already computed, and

499
00:31:03,530 --> 00:31:04,680
what was it?

500
00:31:04,680 --> 00:31:06,870
It was that.

501
00:31:06,870 --> 00:31:10,980
So the second factor was minus
x e to the minus x

502
00:31:10,980 --> 00:31:12,930
squared over 2.

503
00:31:12,930 --> 00:31:15,130
So this is--

504
00:31:15,130 --> 00:31:20,030
can I just like remember
this is f dg dx

505
00:31:20,030 --> 00:31:21,210
in the product rule.

506
00:31:21,210 --> 00:31:23,040
And the product, this is--

507
00:31:23,040 --> 00:31:26,330
here is a product
of f times g.

508
00:31:26,330 --> 00:31:36,500
So f times dg dx, and
now I need g times--

509
00:31:36,500 --> 00:31:41,850
this was g, and this is
df dx, or it will be.

510
00:31:41,850 --> 00:31:44,070
What's df dx?

511
00:31:44,070 --> 00:31:46,970
Phooey on this old example.

512
00:31:46,970 --> 00:31:48,220
Gone.

513
00:31:49,650 --> 00:31:55,880
OK, df dx, well, f is minus
x. df dx is just minus 1.

514
00:31:55,880 --> 00:31:56,980
Simple.

515
00:31:56,980 --> 00:32:00,230
All right, put the
pieces together.

516
00:32:00,230 --> 00:32:04,370
We have, as I expected we would,
everything has this

517
00:32:04,370 --> 00:32:08,490
factor e to the minus
x squared over 2.

518
00:32:08,490 --> 00:32:11,780
That's controlling everything,
but the question is what's

519
00:32:11,780 --> 00:32:16,330
it-- so here we have a minus
1; is that right?

520
00:32:16,330 --> 00:32:19,240
And here, we have a
plus x squared.

521
00:32:19,240 --> 00:32:25,050
So I think we have x squared
minus 1 times that.

522
00:32:25,050 --> 00:32:35,504
OK, so we computed a
second derivative.

523
00:32:35,504 --> 00:32:36,754
Ha!

524
00:32:40,380 --> 00:32:42,990
Two things I want to do,
one with this example.

525
00:32:45,750 --> 00:32:51,630
The second derivative
will switch sign.

526
00:32:51,630 --> 00:32:56,570
If I graph the darn thing--
suppose I tried to graph that?

527
00:32:56,570 --> 00:32:59,900
When x is 0, this thing
is negative.

528
00:32:59,900 --> 00:33:01,530
What is that telling me?

529
00:33:01,530 --> 00:33:03,230
So this is the second group.

530
00:33:03,230 --> 00:33:10,990
This is telling me that
the slope is going

531
00:33:10,990 --> 00:33:13,830
downwards at the start.

532
00:33:13,830 --> 00:33:16,440
I see it.

533
00:33:16,440 --> 00:33:22,140
But then at x equal 1, that
second derivative, because of

534
00:33:22,140 --> 00:33:26,700
this x squared minus 1
factor, is up to 0.

535
00:33:26,700 --> 00:33:30,910
It's going to take time with
this second derivative.

536
00:33:30,910 --> 00:33:33,350
That's the slope of the slope.

537
00:33:33,350 --> 00:33:36,840
That's this point here.

538
00:33:36,840 --> 00:33:38,430
Here is the slope.

539
00:33:38,430 --> 00:33:44,350
Now, at that point,
its slope is 0.

540
00:33:44,350 --> 00:33:48,460
And after that point, its
slope is upwards.

541
00:33:48,460 --> 00:33:50,410
We're getting something
like this.

542
00:33:50,410 --> 00:33:59,470
The slope of the slope, and
it'll go evenly upwards, and

543
00:33:59,470 --> 00:34:00,636
then so on.

544
00:34:00,636 --> 00:34:02,100
Ha!

545
00:34:02,100 --> 00:34:07,730
You see that we've got the
derivative code, the slope,

546
00:34:07,730 --> 00:34:11,110
but we've got a little more
thinking to do for the slope

547
00:34:11,110 --> 00:34:16,440
of the slope, the rate of change
of the rate of change.

548
00:34:16,440 --> 00:34:18,870
Then you really have
calculus straight.

549
00:34:18,870 --> 00:34:27,739
And a challenge that I don't
want to try right now would be

550
00:34:27,739 --> 00:34:31,370
what's the chain rule for
the second derivative?

551
00:34:31,370 --> 00:34:32,170
Ha!

552
00:34:32,170 --> 00:34:41,750
I'll leave that as a challenge
for professors who might or

553
00:34:41,750 --> 00:34:43,469
might not be able to do it.

554
00:34:43,469 --> 00:34:49,040
OK, we've introduced the second
derivative here at the

555
00:34:49,040 --> 00:34:51,179
end of a lecture.

556
00:34:51,179 --> 00:34:58,240
The key central idea of the
lecture was the chain rule to

557
00:34:58,240 --> 00:34:59,790
give us that derivative.

558
00:34:59,790 --> 00:35:00,290
Good!

559
00:35:00,290 --> 00:35:02,610
Thank you.

560
00:35:02,610 --> 00:35:04,420
NARRATOR: This has been
a production of MIT

561
00:35:04,420 --> 00:35:06,810
OpenCourseWare and
Gilbert Strang.

562
00:35:06,810 --> 00:35:09,080
Funding for this video was
provided by the Lord

563
00:35:09,080 --> 00:35:10,300
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564
00:35:10,300 --> 00:35:13,430
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565
00:35:13,430 --> 00:35:16,510
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566
00:35:16,510 --> 00:35:18,070
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