of the chain rule.
You know, people often wonder where the name chain rule comes from. I was just wondering about that myself. So is it because it chains you down? Is it like a chain fence? I decided what it is. It's because by using it, you burst the chains of differentiation, and you can differentiate many more functions using it. So when you want to think of the chain rule, just think of that chain there. It lets you burst free.
Let me give you another application of the chain rule. Ready for this one? So I'd like to differentiate the sin(10t). Again, this is the composite of two functions. What's the inside function? Okay, so I think I'll introduce this new notation. x = 10t, and the outside function is the sine. So y = sin x. So now the chain rule says dy/ dt is... Okay, let's see. I take the derivative of the outside function, and what's that? Sine' and we can substitute because we know what sine' is. So I get cosine of whatever, x, and then times what? Now I differentiate the inside function, which is just 10. So I could write this as 10cos of what? 10t, x = 10t. Now, once you get used to this, this middle variable, you don't have to give a name for it. You can just to think about it in your mind without actually writing it down, d/dt (sin(10t)). I'll just do it again without introducing this middle variable explicitly. Think about it. I first of all differentiate the outside function, and I get cosine. But I don't change the thing that I'm plugging into it. It's still x that I'm plugging into it. x is 10t. So let's just write 10t and not worry about the name of that extra variable. If it confuses you, introduce the new variable. And do it carefully and slowly like this. But, quite quickly, I think you'll get to be able to keep that step in your mind. I'm not quite done yet. I haven't differentiated the inside function, the derivative of 10t = 10. So you get, again, the same result. A little short cut that you'll get used to. Really and truly, once you have the chain rule, the world is yours to conquer. It 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 a