 puts you in a very, very powerful position.
Okay, well let's see. What have I covered today? I've talked about product rule, quotient rule, composition. I should tell you something about higher derivatives, as well. So let's do that. This is a simple story. Higher is kind of a strange word. It just means differentiate over and over again. All right, so let's see. If we have a function u or u(x), please allow me to just write it as briefly as u. Well, this is a sort of notational thing. I can differentiate it and get u'. That's a new function. Like if you started with the sine, that's gonna be the cosine. A new function, so I can differentiate it again. And the notation for the differentiating of it again, is u''. So u'' is just u' differentiated again. For example, if u = sin x, so u' = cos(x). Has Professor Gerison talked about what the derivative of cosine is? What is it? Ha, ok so u'' = - sin x. Let me go on. What do you suppose u''' means? I guess it's the derivative of u''. It's called the third derivative. And u'' is called the second derivative. And it's (u')' differentiated again. So to compute u''' in this example, what do I do? I differentiate that again. There's a constant term, - 1, constant factor. That comes out. The derivative of sine is what? Okay, so u''' = - cos x. Let's do it again. Now after a while, you get tired of writing these things. And so maybe I'll use the notation u^(4). That's the fourth derivative. That's u''''. Or it's (u''')' the fourth derivative. And what is that in this example? Okay, the cosine has derivative -sin , like you told me. And that -sin cancels with that sine, and all together, I get sin x. That's pretty bizarre. When I differentiate the function sine of x four times, I get back to the sine of x again. That's the way it is.
Now this notation, prime prime prime prime, and things like that. There are different variants of that notation. For example, that's another notation. Well, you've used the notation du/ dx before. u' could also be denoted du/ dx. I think we've already here, today, used this way of rewriting du/ dx. I think when I was talking about d/dt(uv) and so on, I pulled that d / dt outside and put whatever function you're differentiating over to the right. So that's just a notational switch. It looks good. It looks like good algebra doesn't it? But what it's doing is regarding this notation as an operator. It's something you apply to a function to get a new function. I apply it to the sine function, and I get the cosine function. I apply it to x^2, and I get 2x. This thing here, that symbol, represents an operator, which you apply to a function. And the operator says, take the function and differentiate it. So further notation that people often use, is they give a different name to that operator. And they'll write capital D for it. So this is just using capital D for the symbol d/dx.
So in terms of that notation, let's see. Let's write down what higher derivatives look like. So let's see. That's what u' is. How about u''? Let's write that in terms of the d/dx notation. Well I'm supposed to differentiate u' right? So that's d/dx applied to the function du/ dx. Differentiate the derivative. That's what I've done. Or I could write that as d/dx applied to d/dx applied to u. Just pulling that u outside. So I'm doing d/dx twice. I'm doing that operator twice. I could write that as (d/dx)^2 applied to u. Differentiate twice, and do it to the function u. Or, I can write it as, now this is a strange one. I could also write as like that. It's getting stranger and stranger, isn't it? This is definitely just a kind of abuse of notation. But people will go even further and write (d^2)u/dx^2. So this is the strangest one. This identity quality is the strangest one, because you may think that you're taking d of the quantity x^2. But that's not what's intended. This is not d(x^2). What's intended is the quantity dx^2. In this notation, which is very common, what's intended by the denominator is the quantity dx^2. It's part of this second differentiation operator. So I've written 