Formal definition of partial derivatives

So I've talked about the partial derivative and how you compute it, how you interpret it 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 multivariable input, so x, y, and you know, put I'll just emphasize that it could actually be, you know, a number of other inputs. You could have 100 inputs or something like that.

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. Um, when you have a function that's just one variable, you know, this would be just something simple like 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 df and dx 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 dx here as representing a slight nudge, just a slight nudge in the input value. So this is in the x direction; you've got your x coordinate. Your output f of x is what the y-axis represents here, and then you're thinking of df as being the resulting nudge here, the resulting change to the function.

So when we formalize this, we're going to be thinking of a fraction that's going to represent df over dx. And I'll leave myself some room; you can probably anticipate why, if you know where this is going. Um, and we'll instead of saying dx, I'll say h. So instead of thinking, you know, dx 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.

And now when you actually say what do we mean by the resulting change in f, we should be writing is, 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 just or the original value of the function at that point? So this top part is really what's representing df; you know, this is what's representing df over here. But you don't do this for any actual value of h; you don't do it for any specific nudge. The whole point, 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 multivariable world, very similar story. Um, we can pretty much do the same thing, and we're going to look at our original fraction and just start to formalize what we think of each of these variables as representing. That partial x is still 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 kind of like to draw this out is you think of your entire input space as, you know, the xy plane. If it was more variables, this would be a high dimensional space, and you're thinking of some point, you know, maybe you're thinking of it as a, b. Well, maybe I should specify that, actually, where, you know, 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. I'm just 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 f partial x, sorry, you're changing the x direction, and this is that resulting change to 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, 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—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. Um, except, oh, I mean, 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—actually, let me just move this guy to 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 dx, any specific size of this—really, this is h, you know, 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 I'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 a 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 going to divide something by that nudge. And again, I'm just using the same variable; maybe it would be clear 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 kind of want the standard go-to limiting variable.

But this time, when you're considering what is the resulting change—oh, and again, I always, 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 going to be at a, but the change happens for that second variable. It's going to 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 moved this over and gave myself some room is because we're taking the limit as this h goes to zero. And the way that you think 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, I just always think about this as spelling out what we mean by partial y and partial f and kind of spelling out why it is that Leibniz came up with this notation in the first place. Well, I don't know if Leibniz came up with the partials, but the df dx portion. Um, and this is good to keep in the back of your mind, especially as we introduce new notions, new types of multivariable derivatives like the directional derivative. I think it helps clarify what's really going on in certain contexts. Great, see you next video.