Exact Equations Intuition 1 (proofy)

Chain rule using partial derivatives (not a proof; more intuition).

Now I introduce you to the
concept of exact equations.

And it's just another method for
solving a certain type of

differential equations.

Let me write that down.

Exact equations.

Before I show you what an exact
equation is, I'm just

going to give you a little bit
of the building blocks, just

so that when I later prove it,
or at least give you the

intuition behind it, it doesn't
seem like it's coming

out of the blue.

So let's say I had some function
of x and y, and we'll

call it psi, just because that's
what people tend to use

for these exact equations.

So psi is a function
of x and y.

So you're probably not familiar
with taking the chain

rule onto partial derivatives,
but I'll show it to you now,

and I'll give you a little
intuition, although

I won't prove it.

So if I were to take the
derivative of this with

respect to x, where y is also
function of x, I could also

write this as y-- sorry,
it's not y, psi.

Undo.

So I could also write this as
psi, as x and y, which is a

function of x.

I could write it
just like that.

These are just two
different ways of

writing the same thing.

Now, if I were to take the
derivative of psi with respect

to x-- and these are just the
building blocks-- if I were to

take the derivative of psi with
respect to x, it is equal

to-- this is the chain rule
using partial derivatives.

And I won't prove it, but
I'll give you the

intuition right here.

So this is going to be equal to
the partial derivative of

psi with respect to x plus the
partial derivative of psi with

respect to y times dy dx.

And this is should make a
little bit of intuition.

I'm kind of taking the
derivative with respect to x,

and if you could say, and I know
you can't, because this

partial with respect to
y, and the dy, they're

two different things.

But if these canceled out,
then you'd kind of have

another partial with
respect to x.

And if you were to kind of add
them up, then you would get

the full derivative
with respect to x.

That's not even the intuition,
that's just to show you that

even this should make a little
bit of intuitive sense.

Now the intuition here, let's
just say psi, and psi doesn't

always have to take this form,
but you could use this same

methodology to take psi to
more complex notations.

But let's say that psi, and
I won't write that it's a

function of x and y.

We know that it's a function
of x and y.

Let's say it's equal to some
function of x, we'll call that

f1 of x, times some
function of y.

And let's say there's a bunch
of terms like this.

So there's n terms like this,
plus all the way to the nth

term is the nth function of x
times the nth function of y.

I just defined psi like this
just so I can give you the

intuition that when I use
implicit differentiation on

this, when I take the derivative
of this with

respect to x, I actually
get something that

looks just like that.

So what's the derivative of
psi with respect to x?

And this is just the implicit
differentiation that you

learned, or that you hopefully
learned, in your first

semester calculus course.

That's equal, and we just do
the product rule, right?

So the first expression, you
take the derivative of that

with respect to x.

Well, that's just going to be f1
prime of x times the second

function, well, that's
just g1 of y.

Now you add that to the
derivative of the second

function times the
first function.

So plus f1 of x, that's just the
first function, times the

derivative of the
second function.

Now the derivative of the second
function, it's going to

be this function with
respect to y.

So you could write that
as g1 prime of y.

But of course, we're doing
the chain rule.

So it's that times dy dx.

And you might want to review the
implicit differentiation

videos if that seems a
little bit foreign.

But this right here, what I
just did, this expression

right here, this is the
derivative with

respect to x of this.

And we have n terms like that.

So if we keep adding them, I'll
do them vertically down.

So plus, and then you have a
bunch of them, and the last

one's going to look the
same, it's just the

nth function of x.

So fn prime of x times the
second function, g n of y,

plus the first function, fn of
x, times the derivative of the

second function.

The derivative of the second
function with respect to y is

just g prime of y times dy dx.

It's just a chain rule.

dy dx.

Now, we have two n terms. We
have n terms here, right,

where each term was a f of x
times a g of y, or f1 of x

times g1 of y, and then
all the way to fn of

x times gn of y.

Now for each of those, we got
two of them when we did the

product rule.

If we group the terms, so if
we group all the terms that

don't have a dy dx on them,
what do we get?

If we add all of these, I guess
you could call them on

the left hand side, I'm just
rearranging, it all equals f1

prime of x times g1 of y, plus
f2, g2, all the way to fn

prime, I'm sorry, fn prime
of x, gn of y.

That's just all of these
added up, plus all

of these added up.

All the terms that have
the dy dx in them.

And I'll do them in
a different color.

So all of these terms
are going to be

in a different color.

I'll do it in a different
parentheses.

Plus f1 of x g1 prime of y, and
I'll do the dy dx later,

I'll distribute it out.

Plus, and we have n terms, plus
fn of x gn prime of y,

and then all of these terms
are multiplied by dy dx.

Now, something looks
interesting here.

We originally defined our psi,
up here, as this right here,

but what is this green term?

Well, what we did is we took all
of these individual terms,

and these green terms here are
just taking the derivative

with respect to just x on each
of these terms. Because if you

take the derivative just with
respect to x of this, then the

function of y is just
a constant, right?

If you were to take just a
partial derivative with

respect to x.

So if you took the partial
derivative with respect to x

of this term, you treat a
function of y as a constant.

So the derivative of this would
just be f prime of x, g1

of y, because g1 of y
is just a constant.

And so forth and so on.

All of these green terms you
can view as a partial

derivative of psi with
respect to x.

We just pretended like
y is a constant.

And that same logic, if you
ignore this, if you just look

at this part right here,
what is this?

We took psi, up here, we treated
the functions of x as

a constant, and we just took
the partial derivative with

respect to y.

And that's why the primes
are on all the g's.

And then we multiply
that times dy dx.

So you could write this, this
is equal to-- I'll do this

green-- this green is the same
thing as the partial of psi

with respect to x.

Plus, what's this purple,
this part of the purple?

Let me do it in a different
color, in magenta.

This, right here, is the partial
of psi with respect to

y, and then times dy dx.

So that's essentially all I
wanted to show you right now

in this video, because
I realize I'm almost

running out of time.

That the chain rule, with
respect to one of the

variables, but the second
variable in the function is

also a function of x, the
chain rule is this.

If psi is a function of x and
y, and I would take not a

partial derivative, I would take
the full derivative of

psi with respect to x, it's
equal to the partial of psi

with respect to x, plus the
partial of psi with respect to

y, times dy dx.

If y wasn't a function of x, or
if y if it was independent

of x, than dy dx would be 0.

And this term would be 0, and
then the derivative of psi

with respect to x would be just
the partial of psi with

respect to x.

But anyway, I want you to
just keep this in mind.

And in this video I didn't prove
it, but I hopefully gave

you a little intuition if
I didn't confuse you.

And we're going to use this
property in the next series of

videos to understand exact
equations a little bit more.

I realize that in this video I
just got as far as kind of

giving you an intuition here.

I haven't told you yet what
an exact equation is.

I will see you in
the next video.