p 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.