[MUSIC] Many of the functions that we'd
most like to differentiate are actually
compositions of two different functions.
This happens in the real world, too. I
mean, look, if you change the number of
flowers, that's going to affect say how
many rabbits there are around to you know,
eat those flowers. And if you change them
with rabbits, that'll affect how many
wolves that forest can support. There's
some really concrete examples of this.
Here's a concrete example. Suppose that f
of x is the number of widgets produced
with an investment of x dollars, right?
With, with more money, maybe, you can
build more widgets. Suppose g of n is the
income that you get by selling those n
widgets. What you're probably really
interested in is not exactly how many
widgets you produce. What you'd like to
know is, for a given investment, how much
money are you going to make, right? Well,
that's g of f of x minus your initial
investment, right, g of f of x is how much
money comes in when you sell the widgets
that you produced with your initial
investment of x dollars, right? This
quantity is measuring the profit on an
investment of x dollars in widget
production. We need some framework, some
general picture that let's us understand
how one thing changing affects something
else and how that thing's changing goes on
to affect something else. Specifically, if
I've got some function h which is a
composition of two functions, g of f of x
in this case, I'd like to know something
about the derivative of h. I want to know
how changing x affects f and then how
changing f goes on to affect g. And I'd
like some sort of formula that gives me
that answer, right? I'd like to know the
derivative of h in terms of information
about how x is changing affects f and how
changing the input to g affects g. I want
a formula for the derivative of h in terms
of the derivatives of f and the derivative
of g. This is exactly what the chain rule
does. What the chain rule says, is that
the derivative of the composition is the
derivative of g evaluated at f of x times
the derivative of f evaluated at x.
Sometimes, people have the idea that the
chain rule looks somehow, that you'd
really expect the formula to look very
different. I mean sometimes people think
this formula looks a little bit weird, you
know? I'm composing functions, but now
it's the derivative of g composes just a
function f. What's going on? You might
think that given the fact that the
derivative of a sum is the sum of the
derivatives. You might be tempted to think
that the derivative of a composition
should be the composition of derivatives,
but that's not the case. But the chain
rule really is capturing what happens when
you chain together these changes. So let's
think about this chain rule, the
derivative of g of f of x is g prime f of
x times f prime of x in terms of chaining
together different changes. I'm trying to
calculate is how changing x changes g of f
of x right? This is the derivative of the
composition. What do I know? Well, I know
how changing x will change f of x, right?
This is what the derivative of f is, is,
is measuring, right? The derivative is the
ratio of output change to input change.
Now, in between here, what I have is the
change in f of x will change g of f of x
in some way. This ratio of changes is
really the derivative of g at the point of
f of x. What is the derivative? You plug
in an input to the derivative to ask how
wiggling that input would effect the
output and that's exactly what this ratio
is. I'm asking how will f of x is changing
affect g of f of x, right? That's the
derivative of g at the point that's
wiggling, f of x.
Well, if you think about it, now, if I
just multiply these two things together,
then I get the change in g of f of x
divided by the change in x. This is the
chain rule, right? If I multiply together
g prime f of x and f prime of x, what I'm
left with is exactly what I want, the
derivative of g of f of x. You can see
this pictorially as well. So here, I've
drawn three number lines. On the first
number line, I've drawn x and I imagine x
is the input to f. And on the second
number line, I've drawn f of x and f of x
is now the input to g. And on the last
number line, I've drawn g of f of x. The
essential question answered by the
derivative is how changing x will affect g
of f of x? But since this is a composition
of functions, I'm going to analyze the
effect of changing x and g of f of x in
stages, right? I'm first going to see how
this changing x affect f of x and how f of
x is changing affect g of f of x. So let's
imagine that I change x by a small
quantity. I'm calling that small quantity
h here, h is not a function, just some
small number, the amount by which I'm
wiggling the input. Now, how is the output
affected? Well, that's exactly what the
derivative measures. Right? The derivative
of f at x tells me how wiggling the input
x would affect the output. So f prime of
x, which is the ratio of output change to
input change times an actual input change
gives me a first order approximation of
the output change. So I imagine the output
is changing by about f prime of x times h.
Now, how does that change in value of f of
x affect g? Well, I have to figure out how
wiggling the input to g will affect the
output of g and that depends on where I'm
calculating the derivative. I need to
calculate the derivative of g at the point
f of x, because, f of x is the point
that's doing the wiggling. So, it's the
derivative of g at the point f of x that
tells me how wiggling the input around f
of x would affect the output to g. So it's
that derivative times the amount by which
the input changed, which is this quantity
here, f prime of x times h. And when you
look at it this way, you can see that for
an input change to x of some small amount
h, the output changes by about g prime f
of x times f prime of x as much, which is
exactly what the chain rule is telling me
should be the case. Since this is the
correct rule, that the chain rule really
is the derivative of the outside at the
inside times the derivative fu nction.
Let's try to see a numerical example of
this thing in action. So as a numerical
example let's consider the function g of x
equals x to the 4th power and the function
f of x equals 1 plus x to the 3rd power.
Andm maybe what I want to try to estimate
is g of f of 1.0001, and now,
approximately what is that equal to? Well,
it's not too hard to calculate g of, of 1,
right? What's f of 1? Well, that's 1 plus
1 cubed, well, that's 2. So what's g of 2>
Well, that's 2 to the 4th, well, that's
16. So I know that g of f of 1.0001 is
going to be close to 16. The question is,
how is wiggling the input up to 1.0001
going to affect the output of this
composition of functions? Well, I could do
it in stages, right? That's what the chain
rule's telling me to do. So I could
calculate first the derivative of f at 1.
Right? And the derivative of f is 1 plus
3x squared, so the derivative of f at 1 is
3. And indeed, if I calculate f of 1.0001,
that's about 2.0003 and a bit more. Now, I
want to try to calculate how changing the
input to g will affect the output of g. So
I should calculate the derivative of g and
that's 4x cubed by the power rule, but
where should I evaluate the derivative of
g? Your first temptation is to calculate
the derivative of g at 1, but that is not
a good idea, because you're not wiggling
the input 1 to g. What you're really
should be calculating is the derivative of
g at 2, because it's this 2 that's going
to be wiggling. When you wiggle the input
to f, it's the output to f, f of 1, that's
going to be changing, so you should
calculate the derivative of g there and
what is that? That's 4 times 2 cubed,
that's 4 times 8, that's 32.
So what we're trying to calculate is g of
f of 1.0001 and we know that that's about
g of, well, what's f of 1.0001? It's about
2.0003. So what happens when I wiggle the
input of g from 2 to 2.0003? Well, that
should be about the output of g at 2 which
is 16 plus how much I change the input by,
times the derivative of g at the point
where the wiggli ng is happening, which is
2 and that's 32. And what's 16 plus 0.0003
times 32, that's 16.0096. So g of f of
1.0001 is about 16.0096. And you can see
this 96 just from the chain rule, right?
The relevant thing to calculate is g prime
of f of 1 times f prime of 1, right? This
is going to tell me how wiggling the input
1 affects the output and g prime of f of 1
is 32, f prime of 1 is 3, and 32 times 3
is 96. So, that's the chain rule and it's
going to take some time for the chain rule
to really sink in. But the chain rule is
super important for two very different
reasons. On the one hand, you've ta know
the chain rule just to be able to compute
derivatives. A lot of the functions that
you'll be asked to differentiate are
actually compositions of differentiable
functions, so you'll need to use the chain
rule to finish those derivative
calculations. But on the other hand,
you've gotta know the chain rule just to
understand how chained together changes
work. In the real world, a lot of things
change, and those changing things affect
other things, and those changing things,
then go on to affect yet other things. And
you've got, got understand how those
changes get composed together, in order to
really understand how the real world
works.