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[music] We've seen df/dx and we've seen d/dx.

Â So now, we're going to think about just dx.

Â This sort of object is called a differential.

Â [inaudible] about dx not d/dx, right? We've thought about that little bit

Â already, right? As an operator, this the thing that does

Â the differentiation. This object is what we're going to study

Â right now. It's called a differential.

Â But what does that even mean? Geometrically, these differentials, so dy

Â and dx, represented some change in the linear approximation that I get using the

Â tangent line. So here, I've got some function that I've

Â graphed, y equals f of x. And here's a point, x, f of x.

Â And I've got a tangent line with a curve there.

Â And here's some change in x, which I'm calling dx, and here's some change in y,

Â which I'm calling dy. And if you think about, what is the

Â derivative? Well, I mean the derivative really is dy

Â over dx, right, in some sense, you know. If you take the derivative and multiply it

Â by how much the input changed by, I mean, you don't actually get how much the output

Â changed by but you get an approximation to how much the output changed by and as for

Â this dy here is being used as, this dy is the change in the linear approximation.

Â Okay, here's a statement in words. In words, dy is a change in the

Â linearization, or the linear approximation, or the tangent line

Â approximation, to this function that you're thinking of as y, depending on x.

Â That's what dy is. I should emphasize that your dy's had

Â better include a dx in there somewhere. At different points in your life, you'll

Â develop hm, different ideas about what dx and dy really mean.

Â So, in a Calculus class, in this Calculus class, if you really want to, you can

Â think of dx and dy as infinitesimal quantities, very small quantities.

Â It's difficult to make this sort of thinking entirely rigourous.

Â But that's certainly an okay way to think if you're trying to get some intuition for

Â what these things are representing. Now, later on in your life, when you take

Â some course, say, a differential geometry course, dx will then be given some better

Â foundation. Dx will be revealed to be a differential

Â form or a covector. I mean, you'll have some actual

Â interpretation of dx but for the time being, you know, if you really want to,

Â you can regard these things as infinitesimal quantities.

Â But honestly, I wouldn't worry too much about how to actually think about these

Â things and focus more on how to compute with these objects.

Â The basic trick is that if y is f of x, then dy is the derivative of f dx, and you

Â don't even need necessarily to differentiate f in a lot cases.

Â The differential satisfies many of the same rules or analogous rules as just

Â differentiation satisfies. So, for example, d of u and v, or u plus v

Â I should say, is du plus dv. D of u times v, is a product rule, is du

Â times v plus u dv, right? And there's a quotient rule.

Â I mean, this differential satisfied all the same or let's just say analogous rules

Â to derivatives. Let's take a look at an example.

Â For example maybe y is something here like x squared and, in that case, what's dy?

Â Well, dy is the derivative 2x times dx and yes, this does relate some change in x and

Â change in y in terms of linearizations. But you don't even need necessarily to

Â write it this way. You could if you wanted to write dx

Â squared and then think that there is power rule of a differential just like the usual

Â power rule, which then says that this is 2x dx.

Â So, in general, d of x to the n would be nx to the n minus 1 dx.

Â We can cook up a more complicated example. For example, let's calculate d of x sine

Â x. Well, this is d of a product and I can

Â compute that by using the product rule for differentials, right?

Â What does that tell me? Well, it will be d of the first thing

Â times the second thing plus the first thing times d of the second thing.

Â Now, instead of writing dx times sin x, I could write that as sin x dx.

Â And what's d of sin of x? Well, this is d of a function.

Â And remember, how do I take d of a function?

Â Well, it's the derivative times dx. So, this is x times the derivative, which

Â is cosine x times dx. Now, if I like, I could factor out the dx

Â and I could write this as sin x plus x cos x dx, now, because I haven't done anything

Â new here, right? I mean, I could have just differentiate x

Â sin x but the point is I'm writing it down without ever, you know, taking derivative.

Â I'm sort of using the rule for how to compute differentials, right?

Â Like the differential product rule and the fact that the differential of a function

Â is the derivative dx. I'm being a little bit vague with how to

Â deal with these differentials at this point.

Â But we're going to be seeing more of these differentials in the coming weeks.

Â