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[MUSIC] Many of the functions that we'd
most like to differentiate are actually

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compositions of two different functions.
This happens in the real world, too. I

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mean, look, if you change the number of
flowers, that's going to affect say how

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many rabbits there are around to you know,
eat those flowers. And if you change them

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with rabbits, that'll affect how many
wolves that forest can support. There's

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some really concrete examples of this.
Here's a concrete example. Suppose that f

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of x is the number of widgets produced
with an investment of x dollars, right?

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With, with more money, maybe, you can
build more widgets. Suppose g of n is the

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income that you get by selling those n
widgets. What you're probably really

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interested in is not exactly how many
widgets you produce. What you'd like to

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know is, for a given investment, how much
money are you going to make, right? Well,

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that's g of f of x minus your initial
investment, right, g of f of x is how much

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money comes in when you sell the widgets
that you produced with your initial

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investment of x dollars, right? This
quantity is measuring the profit on an

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investment of x dollars in widget
production. We need some framework, some

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general picture that let's us understand
how one thing changing affects something

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else and how that thing's changing goes on
to affect something else. Specifically, if

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I've got some function h which is a
composition of two functions, g of f of x

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in this case, I'd like to know something
about the derivative of h. I want to know

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how changing x affects f and then how
changing f goes on to affect g. And I'd

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like some sort of formula that gives me
that answer, right? I'd like to know the

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derivative of h in terms of information
about how x is changing affects f and how

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changing the input to g affects g. I want
a formula for the derivative of h in terms

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of the derivatives of f and the derivative
of g. This is exactly what the chain rule

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does. What the chain rule says, is that
the derivative of the composition is the

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derivative of g evaluated at f of x times
the derivative of f evaluated at x.

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Sometimes, people have the idea that the
chain rule looks somehow, that you'd

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really expect the formula to look very
different. I mean sometimes people think

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this formula looks a little bit weird, you
know? I'm composing functions, but now

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it's the derivative of g composes just a
function f. What's going on? You might

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think that given the fact that the
derivative of a sum is the sum of the

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derivatives. You might be tempted to think
that the derivative of a composition

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should be the composition of derivatives,
but that's not the case. But the chain

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rule really is capturing what happens when
you chain together these changes. So let's

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think about this chain rule, the
derivative of g of f of x is g prime f of

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x times f prime of x in terms of chaining
together different changes. I'm trying to

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calculate is how changing x changes g of f
of x right? This is the derivative of the

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composition. What do I know? Well, I know
how changing x will change f of x, right?

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This is what the derivative of f is, is,
is measuring, right? The derivative is the

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ratio of output change to input change.
Now, in between here, what I have is the

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change in f of x will change g of f of x
in some way. This ratio of changes is

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really the derivative of g at the point of
f of x. What is the derivative? You plug

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in an input to the derivative to ask how
wiggling that input would effect the

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output and that's exactly what this ratio
is. I'm asking how will f of x is changing

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affect g of f of x, right? That's the
derivative of g at the point that's

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wiggling, f of x.
Well, if you think about it, now, if I

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just multiply these two things together,
then I get the change in g of f of x

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divided by the change in x. This is the
chain rule, right? If I multiply together

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g prime f of x and f prime of x, what I'm
left with is exactly what I want, the

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derivative of g of f of x. You can see
this pictorially as well. So here, I've

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drawn three number lines. On the first
number line, I've drawn x and I imagine x

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is the input to f. And on the second
number line, I've drawn f of x and f of x

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is now the input to g. And on the last
number line, I've drawn g of f of x. The

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essential question answered by the
derivative is how changing x will affect g

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of f of x? But since this is a composition
of functions, I'm going to analyze the

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effect of changing x and g of f of x in
stages, right? I'm first going to see how

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this changing x affect f of x and how f of
x is changing affect g of f of x. So let's

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imagine that I change x by a small
quantity. I'm calling that small quantity

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h here, h is not a function, just some
small number, the amount by which I'm

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wiggling the input. Now, how is the output
affected? Well, that's exactly what the

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derivative measures. Right? The derivative
of f at x tells me how wiggling the input

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x would affect the output. So f prime of
x, which is the ratio of output change to

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input change times an actual input change
gives me a first order approximation of

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the output change. So I imagine the output
is changing by about f prime of x times h.

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Now, how does that change in value of f of
x affect g? Well, I have to figure out how

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wiggling the input to g will affect the
output of g and that depends on where I'm

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calculating the derivative. I need to
calculate the derivative of g at the point

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f of x, because, f of x is the point
that's doing the wiggling. So, it's the

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derivative of g at the point f of x that
tells me how wiggling the input around f

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of x would affect the output to g. So it's
that derivative times the amount by which

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the input changed, which is this quantity
here, f prime of x times h. And when you

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look at it this way, you can see that for
an input change to x of some small amount

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h, the output changes by about g prime f
of x times f prime of x as much, which is

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exactly what the chain rule is telling me
should be the case. Since this is the

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correct rule, that the chain rule really
is the derivative of the outside at the

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inside times the derivative fu nction.
Let's try to see a numerical example of

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this thing in action. So as a numerical
example let's consider the function g of x

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equals x to the 4th power and the function
f of x equals 1 plus x to the 3rd power.

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Andm maybe what I want to try to estimate
is g of f of 1.0001, and now,

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approximately what is that equal to? Well,
it's not too hard to calculate g of, of 1,

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right? What's f of 1? Well, that's 1 plus
1 cubed, well, that's 2. So what's g of 2>

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Well, that's 2 to the 4th, well, that's
16. So I know that g of f of 1.0001 is

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going to be close to 16. The question is,
how is wiggling the input up to 1.0001

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going to affect the output of this
composition of functions? Well, I could do

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it in stages, right? That's what the chain
rule's telling me to do. So I could

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calculate first the derivative of f at 1.
Right? And the derivative of f is 1 plus

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3x squared, so the derivative of f at 1 is
3. And indeed, if I calculate f of 1.0001,

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that's about 2.0003 and a bit more. Now, I
want to try to calculate how changing the

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input to g will affect the output of g. So
I should calculate the derivative of g and

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that's 4x cubed by the power rule, but
where should I evaluate the derivative of

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g? Your first temptation is to calculate
the derivative of g at 1, but that is not

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a good idea, because you're not wiggling
the input 1 to g. What you're really

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should be calculating is the derivative of
g at 2, because it's this 2 that's going

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to be wiggling. When you wiggle the input
to f, it's the output to f, f of 1, that's

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going to be changing, so you should
calculate the derivative of g there and

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what is that? That's 4 times 2 cubed,
that's 4 times 8, that's 32.

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So what we're trying to calculate is g of
f of 1.0001 and we know that that's about

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g of, well, what's f of 1.0001? It's about
2.0003. So what happens when I wiggle the

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input of g from 2 to 2.0003? Well, that
should be about the output of g at 2 which

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is 16 plus how much I change the input by,
times the derivative of g at the point

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where the wiggli ng is happening, which is
2 and that's 32. And what's 16 plus 0.0003

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times 32, that's 16.0096. So g of f of
1.0001 is about 16.0096. And you can see

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this 96 just from the chain rule, right?
The relevant thing to calculate is g prime

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of f of 1 times f prime of 1, right? This
is going to tell me how wiggling the input

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1 affects the output and g prime of f of 1
is 32, f prime of 1 is 3, and 32 times 3

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is 96. So, that's the chain rule and it's
going to take some time for the chain rule

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to really sink in. But the chain rule is
super important for two very different

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reasons. On the one hand, you've ta know
the chain rule just to be able to compute

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derivatives. A lot of the functions that
you'll be asked to differentiate are

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actually compositions of differentiable
functions, so you'll need to use the chain

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rule to finish those derivative
calculations. But on the other hand,

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you've gotta know the chain rule just to
understand how chained together changes

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work. In the real world, a lot of things
change, and those changing things affect

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other things, and those changing things,
then go on to affect yet other things. And

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you've got, got understand how those
changes get composed together, in order to

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really understand how the real world
works.