[music] We've seen df/dx and we've seen d/dx.
So now, we're going to think about just dx.
This sort of object is called a differential.
[inaudible] about dx not d/dx, right? We've thought about that little bit
already, right? As an operator, this the thing that does
the differentiation. This object is what we're going to study
right now. It's called a differential.
But what does that even mean? Geometrically, these differentials, so dy
and dx, represented some change in the linear approximation that I get using the
tangent line. So here, I've got some function that I've
graphed, y equals f of x. And here's a point, x, f of x.
And I've got a tangent line with a curve there.
And here's some change in x, which I'm calling dx, and here's some change in y,
which I'm calling dy. And if you think about, what is the
derivative? Well, I mean the derivative really is dy
over dx, right, in some sense, you know. If you take the derivative and multiply it
by how much the input changed by, I mean, you don't actually get how much the output
changed by but you get an approximation to how much the output changed by and as for
this dy here is being used as, this dy is the change in the linear approximation.
Okay, here's a statement in words. In words, dy is a change in the
linearization, or the linear approximation, or the tangent line
approximation, to this function that you're thinking of as y, depending on x.
That's what dy is. I should emphasize that your dy's had
better include a dx in there somewhere. At different points in your life, you'll
develop hm, different ideas about what dx and dy really mean.
So, in a Calculus class, in this Calculus class, if you really want to, you can
think of dx and dy as infinitesimal quantities, very small quantities.
It's difficult to make this sort of thinking entirely rigourous.
But that's certainly an okay way to think if you're trying to get some intuition for
what these things are representing. Now, later on in your life, when you take
some course, say, a differential geometry course, dx will then be given some better
foundation. Dx will be revealed to be a differential
form or a covector. I mean, you'll have some actual
interpretation of dx but for the time being, you know, if you really want to,
you can regard these things as infinitesimal quantities.
But honestly, I wouldn't worry too much about how to actually think about these
things and focus more on how to compute with these objects.
The basic trick is that if y is f of x, then dy is the derivative of f dx, and you
don't even need necessarily to differentiate f in a lot cases.
The differential satisfies many of the same rules or analogous rules as just
differentiation satisfies. So, for example, d of u and v, or u plus v
I should say, is du plus dv. D of u times v, is a product rule, is du
times v plus u dv, right? And there's a quotient rule.
I mean, this differential satisfied all the same or let's just say analogous rules
to derivatives. Let's take a look at an example.
For example maybe y is something here like x squared and, in that case, what's dy?
Well, dy is the derivative 2x times dx and yes, this does relate some change in x and
change in y in terms of linearizations. But you don't even need necessarily to
write it this way. You could if you wanted to write dx
squared and then think that there is power rule of a differential just like the usual
power rule, which then says that this is 2x dx.
So, in general, d of x to the n would be nx to the n minus 1 dx.
We can cook up a more complicated example. For example, let's calculate d of x sine
x. Well, this is d of a product and I can
compute that by using the product rule for differentials, right?
What does that tell me? Well, it will be d of the first thing
times the second thing plus the first thing times d of the second thing.
Now, instead of writing dx times sin x, I could write that as sin x dx.
And what's d of sin of x? Well, this is d of a function.
And remember, how do I take d of a function?
Well, it's the derivative times dx. So, this is x times the derivative, which
is cosine x times dx. Now, if I like, I could factor out the dx
and I could write this as sin x plus x cos x dx, now, because I haven't done anything
new here, right? I mean, I could have just differentiate x
sin x but the point is I'm writing it down without ever, you know, taking derivative.
I'm sort of using the rule for how to compute differentials, right?
Like the differential product rule and the fact that the differential of a function
is the derivative dx. I'm being a little bit vague with how to
deal with these differentials at this point.
But we're going to be seeing more of these differentials in the coming weeks.