# Formal definition of partial derivatives

- [Voiceover] So, I've talked about
the partial derivative and how you compute it,
how you interpret in terms of graphs,
but what I'd like to do here is give its formal definition.
So, it's the kind of thing, just to remind you,
that applies to a function that has a multi-variable input.
So, X, Y, and you know, I'll emphasize that
it could actually be a number of other inputs.
You could have 100 inputs, or something
like that, and as with a lot of things here,
I think it's helpful to take a look at
the one dimensional analogy and think about
how we define the derivative, just the ordinary derivative,
when you have a function that's just one variable.
You know, this would be just something simple,
F of X, and you know, if you're thinking
in the back of your mind that it's a function like F of X
equals X squared, and the way to think about
the definition of this is to just actually spell out
how we interpret this D F and D X, and then
slowly start to tighten it up into a formalization.
So, you might be thinking of the graph of this function.
You know, maybe it's some kind of curve, and when you
think of evaluating it at some point, you know,
let's say you're evaluating it at a point A,
you're imagining D X here as representing a slight nudge,
just a slight nudge in the input value.
So, this is in the X direction.
F of X is what the Y axis represents here, and then
you're thinking of D F as being the resulting nudge here,
the resulting change to the function.
So, when we formalize this, we're gonna be thinking of
a fraction that's gonna represent D F over D X, and I'll
leave myself some room.
You can probably anticipate why if you know where
this is going, and instead of saying D X, I'll say H.
So instead of thinking D X is that tiny nudge,
you'll think H, and I'm not sure why H is used necessarily,
but just having some kind of variable that you
think of as getting small, maybe all the other
letters in the alphabet were taken.
Now, when you actually say, what do we mean by
the resulting change in F, we should be writing,
as well, where does it go after you nudge?
So, when you take, you know, from that input point,
plus that nudge, plus that little H,
what's the difference between that and the original
function, or the original value of
the function, at that point?
So, this top part is really what's representing D F.
You know, this is what's representing D F over here,
but you don't do this for any actual value of H.
You don't do it for any specific nudge.
Largely, the whole point of calculus is that you're
considering the limit as H goes to zero of this,
and this is what makes concrete the idea of, you know,
a tiny little nudge or a tiny little resulting change.
It's not that it's any specific one.
You're taking the limit, and you know, you could
get into the formal definition of a limit,
but it gives you room for rigor as soon as
you start writing something like this.
Now, over in the multi-variable world, very similar story.
We can pretty much do the same thing, and we're gonna
look at our original fraction, and just start
to formalize what we think of each of
these variables as representing.
That partial X, still it's common to use the letter H,
just to represent a tiny nudge in
the X direction, and now if we think about
what is that nudge, and here, let me
draw it out, actually.
The way that I kinda like to draw this out is
you think of your entire input space as,
you know, the X Y plane.
If it was more variables, this would be
a high dimensional space, and you're thinking
at some point, you know, maybe you're thinking of it
as A B, or maybe I should specify that,
actually, where we're doing this
at a specific point how you define it.
We're doing this at a very specific point,
A B, and when you're thinking of your tiny little
change in X, you'd be thinking, you know,
a tiny little nudge in the X direction,
a tiny little shift there, and the entire function
maps that input space, whatever it is, to
the real number line.
This is your output space, and you're saying,
hey, how does that tiny nudge influence the output?
I've drawn this diagram a lot, this loose sketch.
I think it's actually a pretty good model,
because once we start thinking of
higher dimensional outputs or things like that,
it's pretty flexible, and you're thinking of this
the X direction, and this is that resulting
change for the function.
But, we go back up here, and we say,
well what does that mean, right?
If H represents that tiny change to your X value,
well then you have to evaluate the function at the point A,
but plus that H, and you're adding it
to the X value, that first component,
just because this is the partial derivative
with respect to X, and the point B
just remains unchanged, right?
So, this is you evaluating it, kind of, at
the new point, and you have to say,
what's the difference between that and the old
evaluation, where it was just at A and B.
And that's it.
That's the formal definition of your partial derivative,
except, oh, the most important part, right?
The most important part, given that
this is calculus is that we're not doing this
for any specific value of H, but we're actually,
let me just move this guy.
Give a little bit of room here.
Yes.
But, we're actually taking the limit
here, limit, as H goes to zero,
and what this means is you're not considering any specific
size of D X, any specific size of this.
Really, this is H, considering the notation up here,
but any size for that partial X.
You're imagining that nudge shrinking
more and more and more, and the resulting change
shrinks more and more and more, and you're
wondering what the ratio between them approaches.
So, that would be the partial derivative
with respect to X, and just for practice,
let's actually write out what the partial derivative
with respect to Y would be.
So, we'll get rid of some of this
one dimensional analogy stuff here.
Don't need that anymore, and let's just think about
what the partial derivative with respect to
a different variable would be.
So, if we were doing it as partial derivative
of F with respect to Y, now we're nudging slightly
in the other direction, right?
We're nudging in the Y direction, and you'd be thinking,
okay, so we're still gonna divide something
by that nudge, and again I'm just using the same variable.
Maybe it would be clearer to write something like
the change in Y, or to go up here and write
something like, you know, the change
in X, and people will do that, but it's less common.
I think people just kinda want
the standard go-to limiting variable.
But, this time when you're considering
what is the resulting change, oh, and again,
I always forget to write in we're evaluating this
at a specific point, at a specific point A B,
and as a result, maybe I'll give myself
a little bit more room here.
So, we're taking this whole thing,
dividing by H, but what is the resulting change in F?
This time you say F, the new value is still gonna be at A,
but the change happens for that second variable.
It's gonna be that B, B plus H.
So, you're adding that nudge to
the Y value, and as before, you subtract off.
You see the difference between that and how you
evaluate it at the original point, and again,
the whole reason I move this over and give myself
some room is because we're taking the limit,
as this H goes to zero, and the way that
It's just that when you change the input
by adding H to the Y value, you're shifting it upwards.
So, again, this is the partial derivative,
the formal definition of the partial derivative.
Looks very similar to the formal definition of
as spelling out what we mean by
partial Y and partial F, and kinda spelling out
why it is that the Leibniz's came up
with this notation in the first place.
Well, I don't know if Leibniz came up with the partials,
but the D F, D X portion, and this is good
to keep in the back of your mind,
especially as we introduce new notions,
new types of multi-variable derivatives,
like the directional derivative.
I think it helps clarify what's really go on
in certain contexts.
Great.
See ya next video.