- [Voiceover] So in the last video, I introduced this

multi-variable chain rule and here I want to

explain a loose intuition for why it's true,

why you would expect something like this to happen.

So the way you think about an expression like this,

you have this multi-variable function

f of xy and you're plugging things in,

but just that function itself, you'll be

thinking of taking a two dimensional space

you know here's our xy plane,

and then mapping it to, you know, just a real number line

and I'll think of this as f, as the output.

So somehow our whole function takes things from

this two dimensional space and plugs it onto this output.

T you're thinking of just another

number line up here, so t, and then

you've got separate functions here,

you know x of t and y of t.

X of t and y of t.

Each of which take that same value for

a specific input, you know it's not that

they're acting on different inputs,

x of some other input t and y of some other input,

it's the same one and then they move that

somewhere to this output space

which itself get's moved over.

And in this way you're thinking of it

as just a single variable function

that goes from t and ultimately outputs f

it's just that there's multi-dimensional stuff

happening in between and now if we start

thinking about the derivative of it -

what does that mean, what does that mean for the

conception of the picture that we have going on here?

Well, that bottom part, that dt

you're thinking of as a tiny change to t, right?

So you're thinking of it as kind of a nudge,

I'll draw it as a sizable line here

for like moving from some original input over,

but you might in principal think of it as a

very, very tiny nudge in t.

And over here you'd say well, that's gonna move

your intermediary output in the xy plane

to, you know maybe it'll move it in some amount,

again imagine this is a very small nudge,

I'm going to give it some size here

just so I can write into it and

then whatever that nudge in the output space

right, it's a nudge in some direction

that's going to correspond to some change in f.

Some change based on the differential properties

of the multi-variable function itself.

And if we think about this, this change

you might break it into components

and say this shift here has some kind of dx,

some kind of shift in the x direction

and some kind of dy, some shift in the y direction.

But you can actually reason about what these should be

coz it's not just an arbitrary change in x

or an arbitrary change in y,

it's the one that was caused by dt.

So if I go over here, I might say that dx

is caused by that dt and the whole meaning

of the derivative, the whole meaning

of the single variable derivative

would be that when we take dx dt,

this is the factor that tells us, you know,

a tiny nudge in t, how much does that change

the x component and if you want you could

think of this as kind of cancelling out the dts

and you're just left with x, but really you're saying

there's a tiny nudge in t and that results in a

change in x and this derivative is what

tells you the ratio between those sizes.

And similarly, that change in y here,

that change in y is gonna be somehow

proportional to the change in t

and that proportion is given by the

derivative of y with respect to t

that's the whole point of the derivative,

no no, with respect to t and again

you can kind of think of it as if

you're cancelling out the ts and

this is why the fractional writing,

this Leibniz notation is actually pretty helpful.

You know, people will say, oh mathematicians would

like, share their heads at the idea of

treating these like fractions, but not only is it

a useful thing to do coz it is a

helpful mnemonic, it's reflective of what you're

gonna do when you make a very formal argument.

And I think I'll do that in one of the following videos,

I'll describe this in a very, a much more formal way

that's a little bit more airtight than the

kind of hand-waving nudging around.

But the intuition you get from just writing

this is a fraction is basically the scaffolding

for that formal argument, so it's a

fine thing to do, I don't think mathematicians

are shaking their heads every time that a

student or a teacher does this.

But anyway, so this is kind of gives you

what that dx is, what that dy is

and then over here if you're saying

how much does that change the ultimate output of the f?

You could say, well, your nudge of size dx over here,

you're wondering how much that changes the output of f,

that's the meaning of the partial derivative, right.

If we say we have the partial derivative

with respect to x, what that means,

is that if you take a tiny nudge of size x

this is giving you the ratio between that

and the ultimate change to the output that you want.

You could think of it like this partial x

is cancelling out with that dx if you wanted

or you could just say, this is a tiny nudge in x,

this is going to result in some change in f -

I'm not sure what - but the meaning of

the derivative is the ratio between those two

and that's what lets you figure it out.

And similarly, you might call this the change in f

caused by x, like, due to x.

Due to, I should say to dx.

But that's not the only thing changing the value of f right?

That's not the only change happening

in the input space, you also have another change in f

and this one I might say is due to dy.

Due to that tiny shift in y and what that's gonna be

we know it's going to be proportional to that

tiny shift in y and the proportionality constant -

this is the meaning of the partial derivative,

that when you nudge y in some way it

results in some kind of nudge in f and the ratio

between those two is what the derivative gives.

So ultimately, if you put this all together

what you'd say is there's two different things

causing an ultimate change to f.

So if you put these together, and you

want to know what the total change in f is -

so I might go over here and say

the total change in f, one of them is caused

by partial f, partial x - and I can multiply it

by dx here, but really, we know that dx,

the change there was in turn caused by dt

so that in turn is caused by the change

in the x component that was due to dt.

That was of course of size dt.

And then for similar reasons, the other way

that this changes in the y direction

is a partial of f with respect to y

but what caused that initial shift in y,

you'd say that was a shift in y that was due to t,

and that size is dy dt times dt, you could think of it.

So slight nudge in t causes a change in y,

that change in y causes the change in f

and when you add those two together that's

everything that's going on, that's everything

that influences the ultimate change in f.

So then if you take this whole expression

and you divide everything out by dt

so you know, kind of erase it from this side

and put it over here, dt,

this is your multi-variable chain rule,

and of course I've just written the same thing again

but hopefully this gives a little bit on intuition

for how you're composing different nudges

and why you wanna think about it that way.

Of course, you can see this, and you see

the partial f kind of cancels out with that dx

and this partial y kind of cancels out with that dy

and you're left with the two different things

that constitute a change in x,

you know this one is only partially the change in f,

this is also partially the change in f,

but together they give the ultimate change in f

and I think that gives a very strong reason,

if you break it down like that, why this should be true.