a bunch of equalities down here, and the only content to them is that these are all different notations for the same thing. You'll see this notation very commonly. So for instance the third derivative is (d^3)u/dx^3, and so on. Sorry?
Professor: Yes, absolutely. Or an equally good notation is to write the operator (D^3)u. Absolutely. So I guess I should also write over here when I was talking about d^2, the second derivative, another notation is do the operator capital D twice. Let's see an example of how this can be applied. I'll answer this question.
Professor: Yeah, so the question is whether the fourth derivative always gives you the original function back, like what happened here. No. That's very, very special to sines and cosines. All right? And, in fact, let's see an example of that. I'll do a calculation. Let's calculate the nth derivative of x^n. Okay, n is a number, like 1, 2, 3, 4. Here we go. Let's do this. So, let's do this bit by bit. What's the first derivative of x^n? So everybody knows this. I'm just using a new notation, this capital D notation. So it's n x ^ (n -1). Now you know know, by the way, n could be a negative number for that, but for now, for this application, I wanna take n to be 1, 2, 3, and so on; one of those numbers.
Ok, we did one derivative. Let's compute the second derivative of x ^ n. Well there's this n constant that comes out, and then the exponent comes down, and it gets reduced by 1. All right? Should I do one more? D^3 (x^n) = n(n-1). That's the constant from here. Times that exponent, (n - 2), times 1 less, (n - 3) is the new exponent. Well, I keep on going until I come to a new blackboard. Now, I think I'm going to stop when I get to the n minus first derivative, so we can see what's likely to happen. So when I took the third derivative, I had the n minus third power of x. And when I took the second derivative, I had the second power of x. So, I think what'll happen when I have the n minus first derivative is I'll have the first power of x left over. The powers of x keep coming down. And what I've done it n - 1 times, I get the first power. And then I get a big constant out in front here times more and more and more of these smaller and smaller integers that come down. What's the last integer that came down before I got x^1 here? Well, let's see. It's just 2, because this x^1 occurred as the derivative of x^2. And the coefficient in front of that is 2. So that's what you get. The numbers n( n - 1)...2)x^1. And now we can differentiate one more time and calculate what (D^n)(x^n) is. So I get the same number, n(n-1)... and so on and so on, times 2. And then I guess I'll say times 1. Times, what's the derivative of x ^ 1? 1, so times 1. Time 1, times 1. Where this one means the constant function 1. Does anyone know what this number is called? That has a name. It's called n factorial. And it's written n! And we just used an example of mathematical induction. So the end result is (D^n) (x^n) = n! constant. Okay that's a neat fact. Final question for the lecture is what's D^n 1 applied to x ^ n? Ha. Excellent. It's the derivative of a constant. So it's 0.
Okay. Thank you.