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## Multivariable calculus

### Course: Multivariable calculus > Unit 2

Lesson 1: Partial derivatives# Formal definition of partial derivatives

Partial derivatives are formally defined using a limit, much like ordinary derivatives. Created by Grant Sanderson.

## Want to join the conversation?

- 7:07, Is the f(a,b) also being divided by h or is it outside of the fraction?(12 votes)
- The f(a,b) is divided by h. This formula is based on what was covered in1:24to2:46, and is very similar to the df/dx formula he derived. If you remember, the concept of this general formula is centered around dy/dx. dx, of course is h, while dy is equivalent to the numerator in the formula.(19 votes)

- why we do partial differentiation?(6 votes)
- To see how how change in a single variable of a multivariable equation affects said multivariable equation's slope or rate of change. Just started studying multivariable calc but that's my current grasp on the topic.(15 votes)

- at5:30wouldn't this be the partial derivative with respect to a? his notation got a little confusing.. Like I understand that he means that a is a point on the x axis and b is a point on the y axis, but wouldn't it be more proper to have a formal definition just be in terms of f(x,y) because you can take any point on the x and y axis? Or do any of these details even matter?(3 votes)
- He's showing how to get the derivative at any given point using the formal definition. To get a general df/dx and df/dy equation, it's easier to use the method in the section "Partial derivatives, introduction." You can use the formal definition to find a general derivative equation for most functions, but it is much more tedious, especially with higher polynomial functions. Imagine taking the derivative of f(x,y) = x^5 + x^4y + x^3y^2 + x^2y^3 + x y^4 + y^5 so many limits to take for x and y. It is usually taught so students understand better on what derivatives really are.(4 votes)

- imagine i have taken the partial derivative of f(x)=x^2 with respect to y, what will be the graphical expression of this derivative?(1 vote)
- You can't take the partial derivative of f(x)=x^2 because it's a one variable function.(6 votes)

- Why do we call the derivative of a multi-variable function as δF/δx and not just f'(x,y) = why is the term "x" used in the generalized 3-d derivative ?

I understand that to find the derivative of F we would have to compute it in terms of both x and y . Thus i find the term "x" in the general derivative of F confusing.(1 vote)- If we used f'(x, y) to signify the derivative of f, it would be ambiguous as to what we're taking the derivative with respect to - thus we use the x to say that we're taking the derivative with respect to x, and the same with y. I'm not sure where you're seeing that that's the generalized derivative of f, because the real 'generalized derivative' of a function is the gradient, which you'll see later.(4 votes)

- how can this be a formal definition if it does not consider multiple dimensions?(1 vote)
- Because a definition can be formal without being completely generalized, i.e., extended to multiple dimensions.

As an example in another subject area, one can define "integer" without defining "number".(3 votes)

- Why partial x 's resulting change(partial f) is in negative direction?(1 vote)
- He just took it arbitrarily.Depending on the function f ,the change can be in negative or positive direction.(3 votes)

- Hey man, what software do you use to plot this graph and slice it around?(1 vote)
- Do you always need at least two inputs to have a partial derivative? Can you have a partial derivative for one input to the function?(1 vote)
- Technically, you can. However, think about what it would mean.

∂f(a)/∂x = ?

If you apply the formal definition, you get that:

(∂f(a)/∂x) = lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

So, you just get the definition of df(a)/dx.

So, you ∂f(a)/∂x = df(a)/dx.

Because of that, you don't really see people use partial derivatives for a function of a single input, since it's just equal to the normal derivative of that function. But, it's totally possible lol.(1 vote)

- Why isn't that you don't typically see the partial derivatives added together to give the full derivative? You see these added with the chain rule applied to multivariable functions (with intermedieate variables) but NOT to simple multivariable functions.(1 vote)
- as I understand it, adding them won't give us valuable information but will probably give us a useless number because there isn't such a thing as a "full" derivative.

The closest thing to a full derivative is the gradient which allows us to calculate how specific changes in the x and y affect the output f.

The Fx on its own represents how the x variable changes the output of F. (rate of change with respect to x).

The Fy on its own tells us how the y variable changes the output of F as well.

Adding them won't tell us how x and y change the output because x and y can change in different proportions, for example you can change x as twice as much as y, this would make more sense if you think about directional derivatives.

In case of multivariable chain rule, because the variable t changes both x and y at the same time we need to "add" them together to know how t changes f.

I hope I helped.(1 vote)

## Video transcript

- [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. Got your X coordinate, your output, 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 as your partial X, your changing 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 you're thinking about
this is very similar. 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 the derivative, but I just
always think about this 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.