 Let's find the nth derivative of f of x equals sine of x. How do you compute all the higher derivatives of sine? Well, let's go through it step by step. Now, technically speaking, there is such a thing called the zero-th derivative. The zero-th derivative means you've actually taken no derivatives, and so this would be just the original function sine of x. The first derivative we've proven previously is equal to cosine. The derivative of sine is equal to cosine, but we've also computed the derivative of cosine. The derivative of cosine is itself negative sine, which because we know the derivative of sine sticking a negative one in front of it just means we can factor the negative one out by derivative properties, this tells us that the third derivative of sine, which is the derivative of negative sine, would then be negative cosine of x. For which then if we keep on going, we see that the fourth derivative, we're going to keep on going for a while because we're looking for the nth derivative. We want to find a general pattern here. If we take the fourth derivative of sine here, that is the derivative of a negative cosine, taking that negative sine out in front, we're going to get negative times the derivative of cosine, which is a negative sine, but as that's now a double negative, we see that the derivative here is going to be sine of x. Now you'll notice here that I started a new column, that wasn't because I ran out of space, it's because I actually have the same function sine again. If we calculate the fourth derivative, the fifth derivative is actually where we're on right now, if we take the derivative of sine, we're going to get cosine again. And if you're going to take the sixth derivative, that means we have to take the derivative of cosine, which is a negative sine. And if we're going to take the seventh derivative, then you're going to take the derivative of negative sine, which is going to be a negative cosine. And then if you're going to take the eighth derivative, well, you're going to take the derivative of negative cosine, which is again a sine of x. And you see how this pattern keeps on going. If we take the ninth derivative, you're going to end up with a cosine of x. If you take the 10th derivative here, you end up with a negative sine of x. If you take the 11th derivative of sine, you're gonna end up with a negative cosine of x. And if you take the 12th derivative, it's just gonna be sine again. It's gonna start over and over and over again. And so what I would like to do here is try to summarize what we just have seen. If we wanna take the nth derivative of sine of x right here, you're gonna get the following basically four options. You're gonna get sine of x whenever you have a multiple of four, right? So like the zero derivative, the fourth derivative, the eighth derivative, the 12th derivative. Every time you get a multiple of four, you're gonna get back to sine. You're gonna get cosine, cosine of x whenever you're a multiple of four plus one. So like one, five, nine, 13, you're always gonna get the next one down. The next one you would see would be a negative sine of x for which this happens when you get a multiple of four plus two. So that is when you get an even number that's not actually multiple of four, like two, six, 10, 14. You're always gonna get a negative sine of x. And then lastly, you'll get negative cosine of x whenever you're looking at n equals 4k plus three. That is if you're three more than a multiple of four or you can say one less than a multiple of four, you're always gonna get negative cosine. So like three, seven, 11. With those patterns in mind, we can actually predict what these derivatives are gonna be. So if you wanna do something like let's take the derivative, let's take the derivative of f, the 101st derivative. Maybe you're thinking about Dalmatians or something like that. You wanna take the 101st derivative of sine. Well, if you're tagging the derivative 101 times, it turns out that you can ignore any multiples of four because every multiple of four will just give you back a sine. Notice that 101 is actually equal to 100 plus one where 100 is four times 25. So what this tells you is that you're gonna cycle through this, you're gonna go through the cycle sine, cosine negative sine, negative cosine sine, cosine negative sine, negative cosine 25 times. And then you're gonna end up with one. So the 101st derivative is actually the same thing as just the first derivative, which is equal to cosine. So we can calculate the derivatives of sine for arbitrarily large powers. I should mention that cosine's gonna do a very similar statement. It's just it's offset by one. So you can have to change some of these numbers. But again, as you take powers of cosine, it'll go through this cycle over and over again. The derivative of cosine is negative sine. The second derivative is negative cosine. The third derivative is positive sine. The fourth derivative is cosine again. So we can predict higher and higher powers for sine and cosine. So this is actually a very important observation. Notice that sine is its own fourth derivative. Cosine is also its own fourth derivative. We have another function that kinda does something like that. Notice that if you take the derivative of etx, you get back et to the x. The natural exponential is a function for which it's its own first derivative. Now sine and cosine are not quite there. They're a little bit slower than et to the x, but they're their own fourth derivatives. And because of that fact of that observation about derivatives, this actually kinda tells us one of the reasons why sine and cosine are such important functions. They're critical and they're critical tools for solving differential equations because they have these repetitive derivative properties just like et to the x. There'll be some other functions we will see in the future that behave like this, namely the hyperbolic trigonometric functions that is sentient cost, but that's a tale for another day.