So that for this video, we could just do a ton of examples. So let's start off with the first one here. Let's go through F of x = x squared sine x tangent x, and they're all being multiplied together. So this is with three things multiplied together, and we haven't seen three before, but it's a product. So we're probably going to use a product rule and I want you to think of it this way. I want you to think of the first 2x squared inside of the first 2 functions as the sort of first function and then tangent x would be the second function. So whenever you have three things, it's okay, you can always group them moving parentheses around, so if you want like, use parentheses, something like that. You can always add parentheses, so if there were four things you sort of do the same way, five things, whatever. So let's do this. So how do I take the product of two things multiplied by each other? I use the product rule. So the product rule says is the first function x squared sign of x times the derivative of the secant + the second function times the derivative of the first, x squared sign of x. All right, so now I have to find two derivatives as always with any product rule and you might see where this is going. The first derivative I need to find, so I write x squared sign of x and then the derivative tangent. We have this, remember what this is? Secant squared, so times secant is the first term, no problem, + tangent of x times the derivative of x squared times sign of x times the derivative of a product. How do you find the derivative product? Well, we use product rule. And we're going to do that right now. So let's use big friend sees, were going to go first times the derivative of the second function + the second function times the derivative of the first. So, when you have three things multiplied together, you have to use the product rule twice. So it's like saying you have a hammer, you're allowed to hit that nail more than once if you want to. So use the tools that you have as needed and it's okay to use a more than once. So just watch out for your parentheses because if you don't have parentheses, then you're wrong because tangent hits a whole derivative. So here we go, so we have x squared times the derivative sine, we saw that was cosine of x + sine of x times the derivative of x squared, that's 2x. That's my final answer. I would not clean this up in any way, shape or form. Just leave it as is and we're good to go, okay? So that's just one little nontraditional way to see this stuff in action. Let's do another one. Let's do a crazy one. Remember, we said we could take more than one derivatives, we take second derivative, third derivatives. Well, here's what I want. Let's see if you can get use of the notation here. So what is this? What am I asking for? Okay, I want the twenty-seventh derivative of sine of x. And then all of a sudden, I hear the collective groan from the audience, like, my gosh, 27 derivatives, isn't that terrible? Let's try it. There's an expression in math, someone told me this once. I think it's going to device. If you can't do anything smart, do something stupid. So like there's no immediate obvious way to do this other than just hacking at it. The worst thing to do is do nothing is really what they're after. Like the worst thing to do is just sit there kind of stare out the window, panic slowly and be like, well, I don't know what to do. If someone says find the twenty-seventh derivative, let's go off and find twenty-seventh derivative. So, the first derivative also the function is f of sine of x. The derivative of sine is cosine. The second derivative is negative sine, remember derivative cosines negative sine. And, the fourth derivative, now after three, we stop using little tallies. You put the four in the parentheses. The fourth derivative, so the derivative sine is cosine and the negative comes along for the ride, that's negative cosine x. Okay, so like it's not long to do this, we just have a long road ahead of us to try to get all the way to 27. So let's start with the fifth derivative. Let's keep going now. The fifth derivative is the derivative of negative cosine. So the derivative cosine is negative sine and then the negative in front, negative, negative sine becomes positive sine. And all of a sudden, you notice something, say, wait a minute, I just got back to where I started. So if I start taking derivatives again, I have this pattern that's going to emerge, which is interesting. So the seventh derivative is the same as the third derivative, and so on and so on and so on, like they'd repeats in a cycle of length 4. So the eighth derivative is negative cosine and off it goes. And that's interesting. So the twenty-seventh derivative, you think about it is going to be the same as what, like you can subtract and look at the remainder if you want. All the things in the first row, so you think of the function is like the 0 derivative. So, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27. You could just go through the cycle and get there. So, you don't actually have to workout all 27 derivatives if you see the pattern. But of course if you don't start, then you're never going to see the pattern. So if you just work this out and count ' up or use the reminders, you get negative sine of x. So, that's a nice little thing there, and to keep that in mind. So these derivatives there miss math, right, it's beautiful. They have unexpected patterns. If you can't do anything smart, try something stupid. Who knows what stuff just might appear. Let's do another one of these, just for fun. I think they're fun. You may not agree with me, but I do. Let's do another example. Find the derivative of the exponential function, but I don't want the derivative. Let's find the twenty-seventh derivative, you say, okay, I see what's going on. Twenty-seventh derivative, again, this meant to like scare you, the notation, but for all the same reasons, what's the first derivative, e to the x. What's the second derivative e to the x, e to the x. What's derivative of the e to the x, like it just keeps going, right? So hopefully, you're not scared by the questions, just e to x. No problem there, good. Okay, so if they're asking for something crazy, you should expect something to be up. And if you don't immediately see it, just start working it out and look for patterns. Okay, let's do another example. How about we do a combination one maybe? So let's take the function y is 1 + sine of x and will do x + cosine of x. So now I have a quotient. I have trig functions, I have polynomials, nice, big old function here. Find the derivative, put all your rules together. At its heart, this thing is a quotient, so it would become the bottom function times the derivative of the top. Can we do this without having to write driven it up? So what's the derivative? Can we do it together? Derivative of 1 is 0, so I won't write that, + the derivative sine which is cosine. So the derivative of the numerator is cosine -, be careful, be careful, be careful, so bottom touch with top minus the top 1 + sine of x times the derivative of the bottom, that becomes 1- sine of x. All over, the bottom squared, just use parentheses. Just be careful, square the whole thing. Putting all the rules together, this is kind of a straightforward one. Sorry if I disappointed you, but nothing crazy going on here. Just to be careful, and then I would not simplify this. There's really no point to do so. Let's do one more, why not? Again, you'll do a lot more of these for homework for practice, so just to see what they look like, what a just get a couple down with solutions. How about e to the x times cosine of x? This is a product, so we should do the product rule. Let's see is good notation. So f prime of x =, so it's the first function times the derivative of the secant. What's the derivative cosine- sine + the secant function times the derivative of the first, and that's e to the x. Yes, I could factor into the x, but there's no point. I don't want to, so we'll just box this one up and put it under the tree. All right, we could do these all day, we'll do more, so just do lots of practice. Do the odd ones, make sure you can check your answers. And then if anything tricky comes up for like, I'm not sure what they're asking and send me an email. Let me know but we can work those out. Okay, great job on this one. We'll see you in the next video.