Day 1 Calculus

[00:00:00]Okay, so this is going to be day one of the 2-day calculus course and we're going to begin with a really simple idea which is what is calculus. Okay, calculus um is about finding gradients.

[00:00:13]Okay, so if I was to say what is calculus ultimately it is about finding the gradient of a curve. Now if I said what is the gradient of this line over here and I wrote a two and I wrote a -2. What would be the gradient of this line? Would it be easy to find?

[00:00:32]Yes.

[00:00:33]>> Rise over run.

[00:00:33]>> It would be rise over run. The rise would be two units. The run would also be two. So the gradient is one. What's a simpler word for gradient?

[00:00:41]>> Slope.

[00:00:41]>> Yeah. Slope. So I could say this straight line m is equal to one. And that is known as the slope or the gradient. Is that okay? Now for straight lines that's pretty easy. We just do rise over run. But what happens if it gets a little bit more complicated? Give an example. If I was to now sketch something else and I was to sketch let's say this curve over here. Are we okay?

[00:01:00]Uh classic it's a parabola right? Um if I had to make up an equation for this parabola we might say y is equal to x^2 + 1. Are we okay with that? In fact I don't want to use the letter y today.

[00:01:12]I'm actually going to write f ofx a function notation. You guys should all be familiar with this. You did functions in the first term of year 11. Okay. When you look at this um can we find the curve the gradient of the curve very easily? Can we all agree at least it's a little bit harder than a straight line?

[00:01:28]A straight line can we agree that the gradient is a fixed value? It never changes. It's a gradient of one. But the gradient over here is changing all of the time. Let me give you a bit of a trick. If you want to visualize gradients, you want to draw a tangent to the curve. Does anyone know what a tangent to a curve means? Yes. You know the answer. What is it? Um, it's a line that hits one point.

[00:01:49]>> Yeah, it's a line that just touches the curve. So, give me an example. If I was saying today, hey guys, what is the gradient of the curve right at this point? The way you want to visualize it is you want to draw a line at that point. Now, what line exactly do we draw? I can draw what we call a tangent line. Do we see that? Ultimately, it comes back to this bad guy, right?

[00:02:07]Because we only really know how to find gradients of straight lines. And so, the gradient of the curve at that instant is going to be the gradient of this tangent line. Okay, let's do another one. What if I said, "Hey, I want you to find the gradient of this point." Well, the way we do it is we draw a what? What do we draw, guys? A tangent, right? So, if I draw a tangent right at this point over here, well, awesome. We now know that this red line represents the gradient of of the curve at that point. Let's do one more. This guy here, how do you think the tangent line's going to look right at that point?

[00:02:39]It's going to be flat. It's going to be horizontal. And so, maybe it's here.

[00:02:45]Now, do we agree that the tangent lines are continuously changing their gradients? What is the gradient over here? Is that positive or is it negative, guys?

[00:02:53]>> It's negative. Awesome. Great answer.

[00:02:54]And then across over here, what is the gradient? It is zero cuz it's flat. And then over here, what is the gradient?

[00:03:01]It's positive. Great. Now, we have a problem that I want us to solve. And I'm going to teach you the strategy to solve this over the next two days. How the heck am I supposed to find a gradient at every single point on a curve? And the answer is something called the derivative. Okay. Now let me explain it like this. If we have f ofx today f ofx is equal to x^2 + 1. This is called a normal function. Is that okay? This guy has a job. Every function has a job in math. What is the function? What is the job for f? Can anyone tell me what is the job for f ofx? Who knows the answer?

[00:03:34]And if you don't, I can give it to you.

[00:03:35]>> To give you the y value.

[00:03:37]>> Yeah, we grill that in class, didn't we?

[00:03:38]I kept saying it 100 times. f ofx the function f does one job is to give us the y-coordinate. Is that okay? Let me give you an example. If we sub in f of let's say two, then what would this function give us? It will give us 2^2 + 1 which is a value of five. Do we agree?

[00:03:57]What the heck does the five refer to?

[00:03:59]It's saying when x is a value of two, y is a value of five. Do we all understand? That means this point over here has coordinates. I'll just rub that out. 2 comma 5. Are we chill with that?

[00:04:11]So, your function f is going to always give you one thing about the curve. It's going to give you every single y-coordinate as long as you specify what? As long as you specify the x coordinate. You give it the x, it will give you the y. It's a trade. Do we understand? That is f. But today, we're not worried about this guy. You guys did that for one whole term. If you still do not know it, you were not listening in class. What we're going to talk about is fdash of x. Do you guys hear the difference? It's an fdash or f. Yeah.

[00:04:42]Okay. How what does fdash do? Instead of giving you the y-coordinate, it solves the very problem I introduced today.

[00:04:48]What do you think this guy is going to give us? I give it a value of x. What is this guy going to give me in return?

[00:04:54]It's going to give us the gradient of the curve. Is that okay? It's going to tell us is it a negative gradient? Is it a zero gradient? Is it a positive gradient? And also, how steep is it?

[00:05:03]Okay. Now, the derivative of this function. So, every function has a specific gradient function. I'm just going to give it to you for now, and we're going to figure out how to figure it out in a moment. It's actually 2x.

[00:05:15]Now, you're like, "Oh, what? What the hell? Where did that come from?" I'm going to teach you that over the next two days. Just accept it for now, cuz I want you to have an understanding of what this even does. Fash is another function. I'm going to give it a value of x. Let's say fdash of two. Is that okay? If I sub it in, I'll get 2 * 2, which is four. Is that all right? Do you see this red tangent line exactly at x= 2? We now just figured out its exact gradient is four. Not three, not five, it's exactly four. A slope of four. So I can say, let me move this out of the way. I can now label this and I can say the gradient of this m is equal to four. Is that okay?

[00:05:53]Now quick question. What is the gradient at x=0? We already drew it. What's the gradient of the curve at x=0? It is zero. This function tells us exactly that. If I was to sub in fdash of 0, I'll get two lots of zero. And what's 2 * 0, guys? It's zero. So, it's actually behaving exactly as we expect. Let's try one more. Let's say this is -1. If I go fdash of -1 and I sub it into this gradient function today, it gives us 2 * -1, which is -2. Is -2 an expected value here? It is. Do we see it's a negative gradient? Okay, what am I trying to say here? We have f ofx which gives us the y-coordinate of every single point on the curve. Then we have fdash of x which gives us the gradient of the tangent to the curve. Okay, so this gives us the gradient. And if gradient is a really scary word for you guys, hey, I'm chill.

[00:06:49]You can use the word slope. gradient or slope of the curve at a specific point at a specific point or a specific value of x. Do we all understand what f is and what fdash is? Now, over the next hour and a bit, here's what I'm going to do.

[00:07:08]I'm going to teach you every single possible way that I can find f-x if I give you f ofx. So in the exam they'll give you a function f ofx and you're going to have to do some mathematical magic and work out the derivative function or fdash or the gradient. Is that okay? The gradient function. Okay.

[00:07:27]Does anyone know how many ways there are to go from f ofx and I'll write like this f ofx all the way through to fdash of x. Does anyone know how many ways there are to go from one to the other?

[00:07:38]How many? See this guy actually listens in my class so he knows the answers. So there are five ways to get there. I'm going to list out all five and I'm going to teach you all five. So this usually your teacher will take a six weeks to teach you. I'll just teach it to you in an hour and a half. Okay. So there are five ways. Number one is called first principles.

[00:07:59]Number two is known as the power rule.

[00:08:03]Number three is known as the chain rule.

[00:08:08]Number four it is known as the product rule. Number five, it is known as the quotient rule. Now, as you venture into year 12 calculus, you're going to have to learn a few more rules on top of this, but this sets the foundation for us. Okay, so this is the foundation and I'm going to teach you all of these one by one. Okay, the first one I'm going to go through is called first principles.

[00:08:32]First principles.

[00:08:35]Now, unfortunately, because I don't have all of the time in the world, I'm not actually going to show you guys where the first principles formula comes from.

[00:08:45]I'm just going to teach you how to use it. Now, you might be thinking, "Wait, that's disastrous. I don't know where the formula comes from. That means I don't understand maths." Uh, do you guys all know the quadratic formula? I bet half of you don't know where it comes from. And it doesn't matter. What matters is you know how to use the quadratic formula and when to use it. Do you guys understand the difference?

[00:09:02]Here's the formula for first principles.

[00:09:04]Okay?

[00:09:05]um f - x is equal to the limit as h approaches zero f of x + h minus f ofx all over h.

[00:09:17]Remember I said the promise to you guys is I'll teach you how to find fdash of x which is this function over here. Well, this is the first way to do it. And here's a formula. Now when you look at this, do you guys understand there's something that looks really really scary? What looks really scary about this formula other than the fact that it's all letters? Letters are scary. I get it. Anything else that looks really scary about this? Yeah, it's like, oh my god, whoa, pause a second. What the hell did you just write on my board? Limit as h approaches zero. What does that mean?

[00:09:46]I could explain limits if I had infinite time, but today I'm going to keep it simple. When you see this limit notation, it means eventually our goal is to sub in h= 0 into the expression on the right, which is the expression that's inside the brackets over here. It's got a name.

[00:10:06]It's called a difference quotient, but that's lame. So, just whatever. It's that expression. Okay. I want to sub in h equals 0. If I was to do that right now, would I be able to do it? If I sub in h equals 0 right now, what would happen? Can anyone spot the issue?

[00:10:22]>> Yes, because you're going to be dividing by what, guys? You're going to be dividing by zero and that is undefined.

[00:10:27]So, well, that's not good cuz I'm I'm I actually want to find it. Okay, saying it's undefined is not very useful. And so, our goal is to get rid of this in a moment. Now, suppose the question goes like this. Let f ofx is equal, this is the question in in a in a sample question in the exam. Let f ofx= x^2 + 1. Are we all good with that?

[00:10:46]f ofx= x2 + 1. And our goal is to find fdash of x, the derivative function or the gradient function. It's got two different names, doesn't matter. Okay, step number one, we're going to use this formula. But within this formula, there are two things that we need. We need to find f of x plus h. And we also need to find f of x. Do we agree? It's two steps. Step number one, what is f of x plus h? Step number two, what is f of x?

[00:11:10]Let's write down each of these. f of x + h.

[00:11:15]Actually, that's really silly. I should probably make you guys write the easier one first. So, I know that the question already wrote it, but for good practice, I prefer if everyone today could write out f ofx one more time for yourself.

[00:11:27]So, f ofx today is x^2 + 1. Here's the question. What on earth is f of x plus h? Can anyone tell me what that is? If f of x is x^2 + 1, what is f of x plus h?

[00:11:40]You simply want to do what, guys? Yeah, you want to replace every instance of x with x + h. Is that fair? So, it's now going to be x + h^2 + 1. Do we understand that? Now that you've written these two things down, I want you to write down the formula and sub it in for me. So, let's go ahead and do that. f-x limit as h approaches zero f + h - fx all / h. This formula will never change.

[00:12:13]Now, what you're going to do next, guys, is you're going to sub in f of x plus h and f of x individually, one by one. I'm going to make a mistake on the board.

[00:12:20]Okay?

[00:12:22]And I want you guys to tell me what mistake I made. And I did it on purpose, so don't think I'm stupid. All right?

[00:12:28]What mistake did I make other than Eric?

[00:12:31]He's too smart. Can anyone else spot the mistake I made? So, I found each of these and I subbed it into the expression. Looks good. Why is this wrong?

[00:12:42]>> Yes, >> no brackets for the f of x.

[00:12:44]>> Very good answer. You notice that it's minus the entire f ofx. But over here when I did minus, I didn't put brackets around it. So I'm only minusing the first term. That's wrong. You'll get your signs cooked. Put brackets around it. Do you guys understand? That's the first learning point. Make sure you put brackets around the f ofx when you're subbing it in. Step number two, you're going to expand the top. When you expand the top, perfect square. x^2 + 2 hx + h ^2 + 1 - x^2 - 1 all over h. Okay, great. When you expand the top, hopefully your algebra is strong. Can anyone notice what's going to be canceling out, guys?

[00:13:21]Yes. So, you have a positive x squ. You have a negative x squ. What else do we have? We have a positive 1 and a what?

[00:13:27]And a negative 1. So, you're left with not too many terms to deal with. I'm running out of space. And I know some of you guys are right at the back. It's hard to see. So, I'm going to bring it up here. limit as h approaches zero at the top we have 2 hx + h ^2 all over h.

[00:13:43]This is looking really really good and I'll tell you why. At the top, do you notice both of the terms have the letter h?

[00:13:52]If they both have the same thing in math, what are we allowed to do?

[00:13:55]Factoriize. Great. If I factoriize it out of the front Oh, what the hell? Wrong letter. Wrong letter. Back space. Backspace. h of 2x + h all / h. Do you guys remember there was a h at the bottom which was causing us a lot of lot of trouble because it was creating a zero at the bottom. What can we now do with that h expression?

[00:14:15]>> We can cancel it. See you later buddy.

[00:14:18]Right gone. And if we cancel it, we're now left with this.

[00:14:23]Okay, we are finally ready. What did I say? The limit as h approaches zero really simplifies to meaning. What did it really mean at the end of the day?

[00:14:32]Can anyone tell me sub in h= 0? Can we now safely sub in h equals 0? If I do that, take a look at what happens. We get 2x + 0, which of course gives us what value, guys?

[00:14:45]Isn't that exactly what I said was the derivative at the start? We had f ofx is x^2 + 1. The gradient function is 2x.

[00:14:54]Well, we found it. That is the first method of going from function to the derivative. It is known as the first principles. Okay, before we move on, there's a few things I would like to just stress a little bit. Okay, so just listening up over here. Number one, every time you guys do a first principles question, I want you to start with writing two things. What is f ofx?

[00:15:17]And write down what is f of x plus h.

[00:15:20]Number two, when you are doing the question, write down the formula for me.

[00:15:24]Always write down the formula as the first line. Make sure to put brackets around the f ofx.

[00:15:32]And then you're going to expand and factoriize until you have the goal. What is the goal here? To cancel out the what, guys? To cancel out the h from where? From the denominator.

[00:15:47]Once you've been able to achieve that, you can then sub in h= z. There is one last commentary I would love to make.

[00:15:54]Guys, when you're doing this whole thing, please continue to write limit every single line. Your teachers are literally sipping on their tea in their staff room going, "I'm just going to wait until a student forgets to write it and instantly they're going to take a mark off you." Don't fall for that Every single line you have to write limit until when? When did I stop writing my limit, guys? Do you notice at what exact point?

[00:16:19]>> When you sub >> Yeah. When I'm actually subbing it in, then you can drop the limit cuz you're evaluating it now. So when you sub in h equ=0 drop the limit and then you'll get the final answer. Are we chill with that? That is called first principles of calculus of differentiation. Chilling?

[00:16:32]Okay you guys can have a crack at it now.

[00:16:37]Can I get everyone to try example two?

[00:16:39]It does require you to expand a cubic. A little bit scary. Um if you don't know how to expand a cubic, you can expand a perfect square and a linear function.

[00:16:48]Anyway, I won't say too much. I'll let you guys try first. I'm going to need a rub off the board. If you want to take a photo, you can. It's also recorded.

[00:16:54]We'll be posting it on YouTube. You can re-watch it if you want.

[00:16:59]Okay. So, we're going to go through example two now. And here's the idea.

[00:17:03]I've given you f ofx= x cubed. And your goal is to give me the derivative or the gradient function. Step number one. What did I say was step number one? Can anyone remind me?

[00:17:13]>> Yeah. Just write down f ofx again. It's basically probing your brain to not forget that this is the function we are focusing on today. Number two, what did I say? Okay, what should you write guys?

[00:17:22]f of x + h. Now that means every time we see any instance of x, you're going to replace it with x plus h. So that's x + h cubed. Now, some of you might already know how to expand that straight away.

[00:17:36]Some of you might not. If you know how to expand a cubic, go for it. It will look something along the lines of x cub + 3 x2h + 3 xh 2 + h cubed. If you're like, "Where the hell did this guy pull that expression out from?" Then I'll tell you another way as an advanced student how to do it. If you don't know how to expand x plus hub, here's the way. Can we agree that x + h cubed is the same as x + h 2 * x + h? Would that be a fair statement to make? Now, can we expand the x + h 2? Just a squared. What would that give us, guys? That will give us x^2 + 2 hx + x^2 ultiplied by x + h.

[00:18:20]And then through a relatively painful process, if you now multiply this term to both of them and again you can then expand the brackets and simplify to get this. Is that okay? So if you didn't know how to expand a cubic straight away, that's not your fault. They don't really teach it well in schools and you can just use this method. But if you do, then you can write this straight away.

[00:18:38]Let's assume we do to save us a bit of time. What's the next step? So, I've written down f of x and f of x plus h.

[00:18:44]What was the next step I suggested?

[00:18:45]Always give me the what, guys? The formula. Great answer. So, we're going to go f-x is every single line. Do not forget to write the limit h approaches zero f of x + h minus f ofx all divided by h. And I'm just going to go on a journey basically of subbing everything in. I'm going to sub in f of x + h which will give us x cub 3 x^2h 3xh 2 + h cub minus and then I'm going to minus the x cubed. Now because there's only a single term here there's really no need to put a brackets. Okay, cuz there's only one term but otherwise putting a bracket would be the right move. Okay, all divided by h. Okay, awesome guys. Now once you've expanded subbed in and expanded everything, you generally want to see if there's anything that cancels out. Okay, if you look very carefully, can anyone spot what's going to cancel out for us?

[00:19:40]What terms are going to cancel out? The x cubed, right? Can we see that? x cub positive negative. So, they cancel out, which leaves behind 3x^2h 3xh 2 + h cubed all / h. Now, if you get this far into the working out, here's what you want to be looking for. Every single term at the top should have an h.

[00:20:02]And if it doesn't, you screwed up. Okay.

[00:20:04]Do they all have a H at on the top, guys? Yes. Yes. Yes. What do I do with the h's now that they all have it?

[00:20:11]Factoriize. Great answer. So, h outside of 3 x^2 3 xh + h 2 all over h. And now you can really start to see the answer. The h's will cancel out. Does that make sense?

[00:20:25]And then what from here? What's the next step? Who remembers?

[00:20:29]>> Sub in h equals 0. Now, when you sub in h equals 0, you basically nuke these two terms cuz they still have an h. Anything multiplied by 0 is still zero. That would be zero and that would be zero.

[00:20:40]So, what's the only term that's left?

[00:20:43]3x^ 2. Okay? And so, when you sub in h= 0, I'll just write it out for you guys once. So, it'll be + 3x 0 2. These would just turn out to be zero.

[00:20:56]And so, you're left with 3x^2. And that would be the answer. Is that okay? So that is how you do first principles.

[00:21:01]I've done it twice now. Hopefully you guys can work it out on your own. Ask away.

[00:21:05]>> Do we need to do the second last line of work?

[00:21:07]>> Great question. No, you don't. I'm just doing that for the sake of clarity teaching. But uh in a real example, any other questions?

[00:21:16]Okay, great. I'm only going to show you guys one more question in first principles and then we'll move straight on. Okay, this next one might come up in school exams and nobody knows how to do it at the start. Okay, you guys can surprise me maybe. All right, so you guys can copy this down. I'm gonna now go through example. Can you guys go to example four and try it out? All right, I'll give you guys a minute to try it out and then I'll go through it on the board.

[00:21:39]So, what I've done for example four is I've done basically the starting steps that I've always suggested, right? So, write down f ofx, write down f of x plus h. I've written down the formula and I've subbed everything in. Hey, I followed every step that, you know, the tutor guy told me. Uh, and yet I'm still stuck. Yo, that guy sucks. Um, why am I stuck? Because usually what are we able to do at the top? Expand and simplify and then there'll be everything has an h. Is that okay? And you look at that and say nothing has an h and there's a lot of square roots. How do I deal with that? Does anyone have a creative solution to get past this? Oh >> yes. Help me out. What's your suggestion?

[00:22:19]>> Yeah. Yeah.

[00:22:21]>> X as x^ >> Okay, I see. So rewrite that as x^ half and maybe that is x^ half. We could do that and we're still going to be looking at two x to the power of halves and going what do I do from here?

[00:22:34]>> Do you know what the next step might be?

[00:22:37]>> I just thought maybe you could do plus >> okay so like maybe like add another rootx to cancel it or something. Plus minus h plus hus >> oh you mean here >> now you're going to have a minus h under that square root that's going to be stuck. Um, so not going to work. But I love it. You guys want to keep going, okay? Don't feel uh disheartened. So if I did a plus H minus H, there will still be a minus H stuck inside the square root. And that's going to basically stop me from getting any further. I I want to show you guys something. Can everyone look up here on the board? You guys all know the answer. It's deep down inside the depth of your heart. I believe it.

[00:23:11]Because if I said, guys, 1 <unk>x + 2.

[00:23:18]Actually, you know what? I won't even use x. If I said this, what do you reckon a school exam question usually asks you to do from here? Rationalize.

[00:23:26]And how do we rationalize it? We times the top and bottom by what?

[00:23:31]>> The conjugate. And what does the conjugate mean, guys? The conjugate simply means the same expression except change the sign of the second term. Is that okay? Why do we do that? Because that creates a difference of two squares and squares will get rid of the square root. In the same way, you know that you don't only can rationalize the denominator, you can also rationalize the numerator. Let's go ahead and do that. If I times, what do you reckon?

[00:23:56]I'm going to times the top by square<unk> of x + h. And what would be the next symbol, guys?

[00:24:03]Plus roo<unk>x. And at the bottom, you also have to times it so that you are not changing the expression. Okay? So, this whole thing really says times by one. Okay? If you go ahead and do this, guys, take a look at what happens. limit as h approaches zero at the top because it is a difference of two squares a minus b a + b. What would that turn into if we evaluate the numerator?

[00:24:26]>> A square minus b^ 2 which means the first term squared would be x + h. The second term squared will be x. Ah all of the square roots are gone. Now I know what you're thinking. You're like Michael I don't think this is good because all you've done is you just delayed the problem. You move the problem from the top to the bottom.

[00:24:43]Bloody. There's still square root here.

[00:24:46]Trust me. It's okay. We'll work it out.

[00:24:48]Trust me. Trust me. Oh, there's an h.

[00:24:50]So, let me multiply the entire thing by h. Okay, guys. At the top, do you see there's x plus h - x? What would the top evaluate to?

[00:24:59]It would just be h. That's great. And I'll show you guys why.

[00:25:06]What do the h's now finally do? Once again, cancel. Your goal is to always get rid of the h in the denominator. Do you guys understand? You want to get rid of it. And so now we can cancel them.

[00:25:16]And so we're left with limit as h approaches zero 1 /<unk> x + h plus the square roo<unk> of x. Are we chill with that?

[00:25:27]What's the next step from here? Once we cancel out the h, we always do the same thing. Yes.

[00:25:32]>> Great answer. Substitute h equals z. Now watch this guys. I know you're thinking, "Whoa, whoa, Michael, there's still h at the bottom. I thought you said you're not allowed to sub it in. You're only not allowed to sub it in if you sub it in and the entire denominator is zero.

[00:25:46]Is there other brothers and sisters next to it? Now, if I sub in h= 0, do we still have other things in the denominator? So, it is therefore safe.

[00:25:54]Do you guys understand? I can sub in h= 0. If I do that, what would the final expression be? 1 /<unk>x plus what?

[00:26:04]Yeah. So, that would be x plus 0. So, rootx plus rootx. And what would root x plus rootx give us? 2 <unk>x and bummer.

[00:26:11]Okay, that's how you do this question.

[00:26:13]So the technique I'm teaching you is if you need to deal with first principles involving some sort of square root, you can consider doing rationalization, okay, by timesing the top and bottom by the conjugate. This step is super not obvious and if you've never seen it before, you probably wouldn't be able to figure out very naturally, okay? You really have to think deep. Okay, are we all good with that? Fantastic. That is first principles in a nutshell. Okay.

[00:26:40]Number two. Now is the power rule.

[00:26:47]Okay. So if you guys want to follow along in the booklet, all you have to do is keep flipping over until you hit page four and that would be the power rule.

[00:26:54]Okay. Is everyone all chill so far?

[00:26:57]Awesome. Great. All right. Here's how the power rule goes. Easiest thing I've ever taught in my life. There's only two steps. Okay. First principle had like bloody six steps in in in a mix. This is two steps, maybe three if I was to really stretch it, but no more than three. Okay, it's like two steps and maybe three. Uh, step number one, I want you to understand if we have a function which is x to the power of n. Is that okay? The power rule says there are two steps. Okay, step number one, if you want to find the derivative, you are simply going to bring the power down to the front. Guys, what is the power today? n okay great n and then what are we going to do next we're going to copy down the original value of x but remove one from the power so subtract one from the power so it will give us x to the^ of n minus one do we understand that so it's n bring the power down then you minus one from the power there are two steps step one bring down the n and then step two you're going to subtract subtract one from the power. Are we okay with that guys? Now, to give you guys a really trivial example of this, if I can get your attention up on the board, if I have f ofx is x to the power of five, then what would fdash of x be? We're going to bring the power down. So, therefore, we're going to have five and then we're going to minus one from the power. What number would that give us, guys? So, that is your answer. Now I don't know about you but you'll quickly realize that the power rule got us the same result that the first principles formula would have taken 43 years to do.

[00:28:39]So if you had a choice which one would you be using? Power rule. Guess what though? They don't give you a choice in exam. Sometimes they say by using first principles and you're like a okay and then you have to do this to get the three marks. It doesn't matter if you use the power rule to get the answer but you didn't use the first principles formula you get zero. Do you guys understand? So you got to do it. All right, let's continue on. B f ofx is equal to x ^ of 11. My favorite number.

[00:29:05]What do we do from here? If I want the derivative, what's the first thing I do, guys? I bring the power down and then I simply multiply by x to the power of 10.

[00:29:15]Awesome. Do you see how easy this is?

[00:29:17]Who the hell say calculus was hard? It's like numbers like bring down one number and minus one from it. Okay, now let's consider the next case. What if I now said f ofx is 4 * x ^ of 7? H, what do you do when there's a number in front?

[00:29:33]And the answer is bloody nothing. If there is a number that is multiplying the x in front, do not touch it at all.

[00:29:41]Here's what I mean. You're going to write the four as if it was always there. It's there for eternity. You're not going to touch it. You're only going to be differentiating this guy, dx. Do you guys understand? The power rule is affecting x.

[00:29:56]If we now differentiate this, what would I do with the seven class? Bring it down. So 7 * x to the^ of what guys?

[00:30:04]Six. Is that okay? And because Oh, hell no. And then because uh because of these two numbers, they're multiplying. What can I write as the final answer?

[00:30:13]>> 28. So it'll be 28 x to the^ 6. And that's how we deal with numbers in front. So let me ask you the question again. when there is a number that's in front multiplying like the coefficient do I do anything with it in differentiation no leave it there is that okay you're only differentiating the x term and then this is really a simplifying is that okay so 4 * 7 28 okay all right smarties what do we do if I now said f ofx is equal to 5 and I said differentiate it actually you know that's a bit mean I'll do something else first 5x I'm just going to tell you the answer so you guys can commit it to memory. Great.

[00:30:53]You can know how it's just a power rule.

[00:30:54]But at the end of the day, if you have x to the^ of one and you're differentiating it, the x will vanish.

[00:31:01]You're left with just a number. Is that okay? So if I said what is the gradient?

[00:31:06]Can anyone tell me what is the gradient?

[00:31:07]What is the gradient function? It is five. So write this down. Okay. If you just have an x and you're differentiating it, the x will vanish and you're left with just a number. I'm just going to do another one cuz I'm never sure people understand me when I say it the first time.

[00:31:22]1 2 3 4 5x and I differentiate this. What am I left with? -1 2 3 4 5. The number stays. The x vanishes. Is that okay? So your derivative is going to simply be 1 2 3 4 5. Hey, let's do one more for the shits and giggles. What would be pix differentiate it?

[00:31:44]>> It'll be pi. I mean pi is just a number.

[00:31:46]It's just a constant at the end of the day. And this is x to the^ of 1. So when you differentiate this f-x will simply be pi. That's how you deal with x to the^ of 1. Don't even think about it.

[00:31:56]Just drop the x. All right. What about this? Now uh this f ofx= 3. Yes. If you differentiate and there's just the number and x is not even there, then the number vanishes completely and your derivative is simply zero.

[00:32:15]Now some students are really curious like Michael I don't get it now. Why is that true? Can I suggest a way to understand this? First of all answer me class. What does f-x tell you about the curve? What information do we get about the curve?

[00:32:30]>> The the slope or the what guys?

[00:32:33]>> Or the gradient. Yeah, either way is okay. And and then I'm going to take what you said which is f of x= 3. How do we sketch f of x= 3?

[00:32:41]>> It's a straight line. But there are many straight lines in this world. To be more specific, it's a horizontal line. So, can I say that f ofx= 3 is a horizontal line at y= 3? Do we agree with that?

[00:32:52]Well, let me ask you a question. What's the slope?

[00:32:56]>> Ah, do you see that? I haven't changed any of the theory. This is supposed to tell you about the gradient of the original function. If your original function is just a number, that is a horizontal line. And the gradient or slope of a horizontal line is always what number? Hence, this is true. Do you guys understand that? Okay, awesome.

[00:33:18]Now, if we put all of this beautiful stuff together, we'll get a real exam question that might look something like this. F ofx= 3x cub + 5x^2 - 11 x + 2.

[00:33:33]when you have multiple terms polomial just differentiate each of them one by one is that okay so if I was to do fd of x do I need to touch the three at all so I'm going to leave it what would be the derivative of x cubed 3x^2 do I touch the five no what would be the derivative of x^2 >> when you have x to the^ 1 and you differentiate what is left >> just the number when you differentiate a number what is left so don't worry and then we can simplify what's 3 * 3 9 x^2 + 10 x that is oh - 11 and that is your final answer is that okay now I'll say clearly like this this thing that I did I just do it as part of the training wheels cuz we're learning calculus sometimes for the first time in reality you don't have to write this middle line just take the number when you drop it it just multiplies with that see we did it again what would be 3 * three 9 minus one from the power will give us what? X squ. Yeah. What's 2 * 5? 10. Drop one from the power, you just get X. So, you don't have to do the middle line, but I want you to know that that's actually what's happening behind the scenes. And that's why you're just multiplying it with a number in front. Are we all good with that? Fantastic.

[00:34:52]I want everyone to try example five. Get as far as you can and then I'll deal with some of the weird ones like square roots and things like that. Cool. All right. Example five. Do it for me now.

[00:35:03]You guys got it. Uh, can I get everyone's attention now? So, we're going to go through the derivatives of all of these functions. Here's the idea.

[00:35:10]The first few are very easy and then it gets to f and then we get a bit scared.

[00:35:15]So, I'm going to put a little asterisk here for you to like really deep it and focus with me when we get up to there.

[00:35:20]Okay. What is the derivative? Oh my gosh. Why is this guy back? What's the derivative of x^ 7? It'll be 7 x^ of 6.

[00:35:27]Okay. Great. Now, f ofx. Okay. How would I diff this?

[00:35:32]>> 12 x to the power of >> 3 + >> 22x. Great. What would be the derivative over here?

[00:35:40]>> It will be two. Right? When we have x just to the power of 1 and you differentiate it, the number stays the x will vanish. Okay. Over here f dash of x. Let's continue on. What would this be? 6 x^2 - 10 lots of x + 6. The minus 3 is a number. If you differentiate a number alone, it will disappear and vanish. Moving on. F dash of x is equal to oh this one. When you move the three down and you times it by half, what would the fraction become?

[00:36:10]>> 3 over 2x^ 2 and then that would be - 7 and that's the end of it. Is that okay?

[00:36:17]Calculus doesn't have to be difficult.

[00:36:19]There's like two I've only taught you two things and we've been able to just keep on using it again and again. Okay.

[00:36:25]Now, what happens when you have x in the bottom? Listen carefully. The power rule only works when x is in the numerator, not the denominator. So, we're going to have to use our index laws. Is that okay? Looking over here, can I before you differentiate, I want you to write rewrite the function. If there's x squ at the bottom, what can we rewrite it as? 3 * what guys? x to the power what?

[00:36:53]>> Minus 2 index laws. Is that okay? From here, we can now do the derivative.

[00:36:58]Bring the power down. If I'm bringing down a minus two, what would the three turn into, guys?

[00:37:04]-6. And then it'll be x to the power of if we subtract one from the power, it's -3. Don't write negative 1. I do that all the time. So, I'm just learning from experience, too. -3. Is that okay? Now, you could leave your answer like that. I could also give you a pizza box and have the pizza box open and serve it to you.

[00:37:23]That would be pretty dog. Okay. So, yeah, sure. Sure, we could do this, but wouldn't it be so much nicer if we wrote it in the original form that the marker gave it to us in? What's another way of writing x^ minus 3? We can send that guy back to where? Back down to the denominator. So, it'll be -6 / what, guys? x cubed. Okay. And that would be the answer. Oh, what the hell? Okay, that would be the answer. Awesome. Okay.

[00:37:48]Over here, do you notice there's an x at the bottom? The the detrimental mistake is to play around the five. The five is a number. I don't care. The five could be anywhere it wants. It could be floating in space. I don't care. I only care about where the x is. Is that all right? So, when I'm looking at this question, can I please suggest us to write this as 2 over 5 * 1x? Do you guys understand? I don't care about the five being at the bottom. I only care about the x. What can I rewrite the x as?

[00:38:17]>> Can anyone tell me what's another way of writing one of x? Yes. x^ - >> great answer. So it'll be x^ minus1. If we now take the derivative and I bring the minus1 down that would give us -2 over 5 x to the power what guys?

[00:38:35]2 if I minus one from it. Again I could write it like this but I think the better way to do it is to return it to the original form. If I get rid of the negative power where would the x squ go?

[00:38:47]Back down to the denominator. Is that fair enough? And so therefore, I'm running out of space. I'll just write the answer. -2 over 5x^2.

[00:38:57]Okay, only the x cannot be at the bottom. Any other numbers can stay at the bottom. Don't mess with it. It keeps it easier that way. Moving on.

[00:39:05]5 rootx. Does anyone know how to write 5 rootx as a exponent form? X^ half. So, can we perhaps write f ofx as 5 * x to the^ of half? All of a sudden we're back to this particular form and now we can differentiate again. So what would fdash of x be? I bring the half down. What would the five turn into?

[00:39:28]Five on two. And then it'll be x power.

[00:39:30]Now be careful here. What's half minus one? Negative half. Okay? So just be really careful. Now in an exam I'm so cautious of doing a silly mistake. I low-key put really dumb things into the calculator, including half minus one sometimes, just to make sure that I really see a negative half. I put crap like what's 8 plus 3 to make sure it's actually 11, you know, like. Okay, the point is this is the answer. Am I going to leave it like that, do you think? No, we're not. When there's a negative, I can get rid of the negative. Where would I put the x, guys? Back down towards the where? To the denominator. So 2x to the power half. But what's another way of writing x to the power half? Root x. So it'll be even better. Oh my gosh, I'm running out of space again as 5 over 2<unk>x.

[00:40:15]And that would be a great way to write your final answer. Is that chill? Okay, over here. This is combining basically all of the skills I taught you. Is it a problem that there's a two and a seven over there? Do I care about the numbers?

[00:40:28]No, we don't. You know what's the first step? Split it up. -2 over 7 is simply some innocent number at the front. I don't really care about it. 1 over square or cube actually square root of x^ 3. Okay, does anyone know how to turn this guy into a third form?

[00:40:46]Yeah, there's a little two outside for a square root and that goes to the bottom of the fraction. Okay, so this is going to become -2 over 7 1 / x ^ of 3 / 2.

[00:40:58]Where did the two come from? That little two on the outside. We slot it into the denominator. Do we like x at the bottom though? No, we don't. So what can I do next guys?

[00:41:08]X to the power of >> -3 over2. Okay, now I'm going to do the derivative. Do you notice the entire time I'm not differentiating? I'm simply rewriting the original function. So it's just f ofx. When I'm ready to differentiate, I'm now going to write fdash of x. Okay. Now from here, if I bring down the power, negative and negative is going to make a positive.

[00:41:29]The twos will cancel out. you're going to get 3 over 7 * x ^ of if I want to minus one from that does anyone know what immediately what the answer is 5 on 2 great so you take the bottom number you minus it from the top so five on two okay and that would be the answer right no it's okay you could leave it like that but I think we can do better I'm going to put get rid of the negative put it back down at the bottom holy that looks Looks horrible. And then I'm going to write as x to ^ 5 to the square root to really return it back to its original state. Are we all happy with that? Awesome.

[00:42:10]Now, [clears throat] just for the record, when we are doing stuff like example six, it's always just the same rules except you might have to expand the brackets first or split it up and stuff like that. Can you guys all have a go? I'm going to give you guys 2 minutes and then I'm going to speedrun it. All right. So, give you guys 2 minutes and then I'll speedrun it. Can I get everyone's attention here? For this first question, I'm going to show you two ways of dealing with the scenario.

[00:42:37]There is one way that I personally prefer cuz it teaches you a skill that you'll need in year 12 a lot. Okay, but here's what the first way is. Okay, so method number one, you're going to realize that there's x's at the top and at the bottom. It's kind of all over the place. No good. Step one, rewrite the x cubed. If I want to get rid of the x cubed at the bottom and it goes to the top, what would its power turn into? -3. Good answer guys. So it'll be x ^ -3 times that entire numerator. Do we agree with that? Can I now expand it? So multiply it in. So what would that be? If I got x^ -3 * x^2, what do the powers do according to index laws? They add. So it'll be x^ -1.

[00:43:20]Next one + 2 x upon -2 and then -3 x upon minus 3. Do you see how all of a sudden we've returned the function into individual terms containing x? Now I can differentiate. Okay. So the derivative of this would now be -x ^ -2 -4 ^ -3 pos9 -4. Okay. Okay, so I did that a little bit faster than usual just because it is the same steps that we've been doing the entire lesson so far. So I brought the power down, I minus one from the power. Brought the power down, minus one from the power. That's method number one. Before I go further, did everyone understand the essence of that question? So when you see an x at the bottom, what are we going to think of doing now? Moving it to the top and then we can expand it to get individual terms again. Yeah, we always need to get back to this guy because I've only given you one rule. And so if it's not this guy, you literally don't know what to do yet.

[00:44:18]Okay. So So this is good. Um I'm going to show you guys method number two perhaps, which will be I think now normally what teachers like is not what students like. So I know that I like this method. If you don't if you don't like it, it's okay. But it there's a skill that you need to know. So can I get everyone's attention here?

[00:44:38]When you have multiple terms at the top and a single term at the bottom, you can do something called splitting the fraction. Do you agree that all of the terms share the same denominator? So, I can rewrite this as x^2 x cub. Do we agree with that? Then plus what would be the next one?

[00:44:56]>> 2x cubed. And then I'm going to put a minus in between. And then it will be 3x cub. Are we okay with that? Then I can rewrite each of these into index form.

[00:45:06]What's x^2 / x^ 3? What's another way of writing that? X to the^ what guys?

[00:45:12]Does anyone know? If you divide, you want to subtract the powers. Is that okay? Yeah, it will be minus one. Great.

[00:45:18]And it will be + two. If you subtract the powers, what would it be? Minus2.

[00:45:24]And then it'll be minus 3 x^ minus 3.

[00:45:27]And you'll see that that gives us exactly the same three terms as before.

[00:45:31]And then we can differentiate one by one. Now, if you're like, "Oh, Michael, I really hate it out with a passion."

[00:45:36]Hey, that's okay. We don't all have to love the same things. But learn something over here. One of the skills I want to develop is your ability to say split the fraction. Later in year 12, when you learn a topic called integration, we're going to be splitting up a lot of different fractions. So, it's important you get comfortable with that concept. All right, [snorts] taking a look over here. Do you notice that it's basically the same question?

[00:46:01]What can I rewrite the square root of x as? Guys, can anyone suggest another way of writing the square root of x? It will be x to the power of what? Half. Great answer. Do you see there are the same two ways? I could either move that to the top with the negative power and expand or I could split the fraction.

[00:46:19]Now, cuz I get control of the whiteboard, I'm going to split the fraction. [snorts] So, we have 3x over x to the^ of half minus 5 overx to the power half. And then all we have to do is subtract the powers. Do we agree?

[00:46:33]Because they're dividing. What is 1 minus half, guys?

[00:46:39]Half. Yes. Not a trick question. Jesus.

[00:46:42]And then if we move that up, it'll be 5 x^gative half. Okay. So there's no x's on top. So we just add a negative to the power. Okay. And then we can start to differentiate this guy. So remember derivative only has two steps. I'm gonna say it again. Oh, what happened to this from the power?

[00:46:58]You're going to bring down the power. So it'll be 3 over two and then you're going to minus one from the power.

[00:47:04]You're going to bring down the power and then you're going to minus one from the power. Again, I could clean this up and make it look better, but for the sake of time, you guys get the idea. I'll just leave it there. That is technically the answer.

[00:47:16]This guy, this way easier. What do you reckon we have to do first to differentiate this?

[00:47:21]Anyone have a suggestion? There's x outside, there's x on the inside. What a mess. What do you reckon I should do?

[00:47:27]expand. Can I suggest writing f ofx? If we multiply it into each of the terms 2 x ^ 4 - 5x cubed, are we okay? And then we can now differentiate once again.

[00:47:39]fdash of x would be 8 x cub - 15x^2.

[00:47:44]Every single time I am bringing down the power minusing one. Bless you. Bringing down a power minusing one. Okay?

[00:47:52]Literally I've only taught you like one rule. Okay? The power rule. [snorts] Okay? Uh, Eric, your time to shine. What do I do with this? Yeah, you expand it.

[00:48:01]Can we see that this is actually a what?

[00:48:05]>> Yeah, it's a + b a minus b. So, if I was to expand it, it will be a 2 minus b^ 2, which will now be x^2 - 1x^2. Uh-oh, I don't like the x at the bottom. What can I do to solve that issue?

[00:48:23]bring it up to the top by writing -2 power. Yes. So can we now write as x^2 - x^ -2 and then I can get the derivative.

[00:48:32]Guys, what is the derivative of x^2?

[00:48:34]>> 2x. If I differentiate this and I bring the power down, be careful of the negatives. Negative will be what? 2x^ of -3. Okay? Don't say negative 1 because you're minusing one from the power. All right, last one. What the hell? Why is there trig? Yeah, it's just a bit of fun guys like yo sin square plus cos square.

[00:48:55]If you know your Pythagorean identities from trig, what is sin square plus cos square?

[00:49:00]>> It's one. So this is just a troll question. The question is just f ofx is equal to one. Yeah, we love that. What's the derivative of one?

[00:49:09]>> Zero. If you differentiate any number, I'll just finish it off. The derivative of any number is zero. Okay, good. We learned so far in day one we learned uh that f ofx is a function that tells us the y-coordinates. fdash of x is another function but it tells us the slope of the curve and there are five ways to get from one to the other. We have already learned the first principles formula and we have just covered the power rule. The next thing is the chain rule. Now what is the chain rule? Um in order to explain its relevance I might just give you guys an example. If I said f ofx today is equal to 2x + 1^2 is that okay?

[00:49:49]Uh how would I differentiate this? Well so far the method that we'll take is we'll expand it. Is that okay? If I was to expand a perfect square, what would this give us guys? 4x^2 + 4x + 1. Would that be fair enough? And then I can differentiate it, right? Because I've got each of the terms in the form x ^ n.

[00:50:07]What would be the derivative of this?

[00:50:08]Just really quickly, guys, it'll be 8 what guys? 8 x and then that would just be + 4. Hey, no big deal. We're good.

[00:50:17]Okay, what if I said um f ofx is 2x + 1 to the^ of 3? And you might go, well, I know it's a bit of pain, but hey, Michael, I still got it. I could expand it and I could differentiate it.

[00:50:29]I'm like, great. What if I do this?

[00:50:32]Yeah, good luck, right? I mean, in theory, you could expand this. It will take you until next Sunday, but you'll get there. And then after the expanding, you're going to have to differentiate each of them one by one. And God knows there'll be there'll be a lot of terms.

[00:50:46]Uh not fun. So how are we going to do this? The chain rule, listening carefully, is used when there is some function of x that's raised to a diabolical power. Is that okay? [snorts] Um so you know squared is fine. I could expand it, but but 30 that's horrible.

[00:51:02]So here's how we're going to do it. If I can get everyone's attention. Suppose we have uh f actually I'll do it like this. Oh, okay. Okay. I might regret this notation. Instead of using f for the function today because I've already used it here, I'm going to use g of x. Okay.

[00:51:18]It's just another function of x. Um suppose we have g of x. So some function of x raised to the power of n. Do you guys understand that? To differentiate this. So to find a derivative, we're going to actually use the power rule.

[00:51:33]What's the power rule? Say what's the first or the power rule? Class bring down the power. So it will be n then it will be g of x to the power of what guys >> n minus one. Is that okay? Except there's one extra step which is you have to then multiply that by the derivative of the inside function. Okay. What is the derivative of g of x chord? It will simply be written as g >> dash of x. And that would be the formula. Okay. So really the way I like to memorize it in very simple terms is it is the power rule plus a little bit more.

[00:52:10]The chain rule is nothing but plus nothing but the power rule plus a little bit more. Can we understand that? Okay, let's go ahead and try this. If I gave you f ofx is 2x + 1 to the power of 30.

[00:52:22]What would be the derivative? What's the first step? The power rule. If I bring down the 30 and then I rewrite everything, what would be the new power, guys?

[00:52:34]>> 29. Am I done? Not yet. It's the power rule plus a little bit more multiply.

[00:52:40]What is the derivative of the inside?

[00:52:42]So, I want you to look inside the brackets. What is the derivative of 2x + 1? So, therefore, that would be your answer. Is that okay? And of course, what's 30 * 2?

[00:52:52]it will be 62x + 1 to the power of 29.

[00:52:56]Do you guys understand the chain rule is when you have a function raised to some power. Okay, let's do one more example of that before we move on. So if I said f of x today x^2 + 3x - 1 ^ of 15.

[00:53:12]Can anyone please help me out? How do I differentiate this? You'll notice there's a function of x raised to a diabolical power. So we're therefore going to use the chain rule. Bring the power down. What would be the new power, guys?

[00:53:26]>> 14.

[00:53:27]>> 14. Am I done? Absolutely not. You need to remember the little bit more. Which is times by the derivative of whatever's on the inside. What is the derivative of x^2 + 3x - 1?

[00:53:39]>> 2x + 3. Now, I'm going to write it wrong. This is wrong. Why?

[00:53:46]>> No brackets. If you do not have the brackets, this is only multiplying the 2x, not the entire expression. So please remember to have your brackets over here. Are we all understood? Okay, great. Now suppose I give you a more interesting scenario. Oh, bless you. If we have f ofx is equal to 1 / uh x + 1 to the^ of 3. Hm. Do you guys notice this is very similar to what we were doing before? We have an x, but it's now at the bottom. What do you reckon we're going to do? bring it to the top. If I bring it to the top, what would it now read? X + 1 to the^ minus 3. Do we understand? And then from here, I can now differentiate using the chain rule.

[00:54:28]There's a function of X raised to some weird diabolical power. First step, power rule. Bring down the power minus 3. X + one to the power of what, guys?

[00:54:40]>> H another two. Yeah. Yeah. -4. And then we need to multiply by the derivative of the inside. What is the derivative of x plus one?

[00:54:48]>> Yes. So, whatever. Yeah. Anything times one is just one. So, not that I didn't do the step. You know what? Just in case you guys at me here, I did do it, but it's just multiplying by one doesn't make a difference. So, you don't have to write that step. Cool. All right, smarties. What about this then? What if I say 3 over 2 to the square<unk> of x^2 - 1? Oh, what are we going to do now?

[00:55:12]Does anyone have a suggestion on step number one to clean this up? cuz this this looks horrible right now.

[00:55:18]Do I need to worry about the three and the two? Remember, I don't care about the numbers. I care about the x's. So, the first thing I suggested is write 3 over2 * 1 / the<unk> x - 1. It should look like that. Do I like square roots?

[00:55:33]Absolutely not. So, can I rewrite it as 3 over 2 * 1 / x^ 2 - 1 to the^ of half?

[00:55:40]Is that fair enough? Do I like x at the bottom? No. So we can bring it to the top and write it as what power.

[00:55:48]So we're going to be left with 3 / 2 * x^2 - 1 to the^ of half.

[00:55:55]Have I differentiated yet? No. This was just me rewriting the function so that it's presented in a way that is differentiable. So from here we can say fdash of x is equal to step one power rule. Bring down the power. Can anyone tell me what the number would turn into if I brought down the power right now?

[00:56:15]-3 over4 is the right answer. Awesome.

[00:56:16]-3 over4 it will be x^2us 1. If I subtract one from the power it will be minus 3 over2. Am I done? No. There's a little bit more. What is the little bit more? Derivative of the what?

[00:56:31]Times 2x. Is that okay? And of course we can simplify this but I'm just going to leave it there because I just want to show you the strategy. You guys got it?

[00:56:39]That is the chain rule. It is the power rule plus a little bit more. Okay, I will proceed to shut up and let you guys cook. So, can we go to example seven and eight and you guys can start working on it now for me? Let's go. Okay, so now that we've had a chance to try it, I'm going to go through examples 7, 8, 9, all the way through to 10. I'm really excited about teaching you guys these two cuz there's an application. So, finally, we're not just looking at differentiating blindlessly. We're using it for some purpose. Okay, so if I can get everyone's attention here, let's begin with the basics over here. Okay, for the record, can I get everyone to look here for a second? If I give you f ofx, then what is the gradient called? What do I write? F what? fdash of x. If I give you y is equal to the derivative is called dy over dx. It's just notation.

[00:57:32]You don't have to worry about do this too much. Just know that fdash is the same as saying dy dx. Okay, if you're like, "What the hell is dy dx?" It's really just rise over run. Do you see?

[00:57:44]Rise is the change in the y value and run is the change in the x value. So the gradient function rise over run dy dx.

[00:57:52]Now it's more technical than that, but it's beyond what we give a crap about today. So just write it. Okay. So for this function here, if I want the derivative dy on dx, the first derivative, what would I do? What rule do I need to use? I want to hear it first. What rule?

[00:58:05]>> Power rule. Power rule says what? Bring the power >> down. And then it'll be 2x + 1 to the power of 7 multiplied by the derivative of the inside which is what?

[00:58:15]>> Two. So this gives you 16 2x + 1 ^ 7 all together. That is the chain rule. So power rule plus a little bit more. Let's continue here. dy over dx it's equal to if I bring the power down what would I get? 10 x^2 + 3 ^ 4* what is the derivative of the inside? 2x. So be really careful there. It's 2x. So 10 * 2x will be 20x x^2 + 3 ^ 4. Again, dy on dx is equal to bring the five down.

[00:58:50]That's the power rule. And a little bit more. What is the derivative of the inside? Six. Lots of x. Of course, they multiply together to give you 30x 3x^2 - 1 to the^ 4. Okay.

[00:59:04]Can I get everyone's attention here?

[00:59:06]Watch this. Watch this. Do we agree if we look far ahead into the future, I'm going to need to differentiate the inside at some point? And that means I need to differentiate the square root of x. Let's figure that out first. And then I'm going to teach you a hack to memorize this. So if I have f ofx equals the roo<unk> of x and I want to differentiate it. What should I rewrite the rootx as? X^ half. What would be the derivative? I bring down the power. Yes.

[00:59:31]So it will be half time what guys ^ minus half. If we have a negative half that simply means it is a square root.

[00:59:40]But where is it? At the denominator. So this is the same as saying 1 / 2<unk>x.

[00:59:46]Is that okay? Now lock in. Lock in.

[00:59:49]Listen listen. Listen.

[00:59:51]Whenever you see a square root and you want to differentiate it, can I get you guys to commit to memory? It just becomes 1 over 2<unk>x. You're going to use this so much throughout the course, you might as well memorize this specific case because it'll save you so much time. So, I'll say it again. If you have square root of x and you want to differentiate it, it becomes 1 / 2<unk>x. Memorize this one for me. Okay?

[01:00:15]Because if you can commit that to memory, it'll speed up your entire process. Watch this. If I want to now differentiate this guy, let's bring it down here. dy over dx equals. What is the first rule we're going to use, guys?

[01:00:29]Power rule. 7<unk>x - 1 ^ 6. Is that okay? Oh, I need to differentiate the inside. I already know what it is cuz I've been told to memorize it. Do you see how much faster it is if you just know it? Uh, now from here, this expression, when it's a multiplier, guys, does it go to the top of the fraction or the bottom? Great answer.

[01:00:50]the top 7 roo<unk>x - 1 ^ 6 over 2<unk>x so much faster that way. So if you guys can really promise me to memorize this identity, it will be really great for us. Okay, moving on over here. I do not like the x + one at the bottom. So what can I rewrite it as?

[01:01:08]One power. Yes, we've done it again and again. dy on dx is equal to bring the power down minus one from the power. What is the derivative of the inside?

[01:01:22]What's the derivative of x plus one?

[01:01:24]Yeah, it's one. So therefore, I don't need to do it. Well, I mean, look, I'll write it for today. And it'll be - 1 / x + 1^ 2. Awesome. Okay. Same thing over here, isn't it? If I want to differentiate this, Eric, what is the first problem I need to solve?

[01:01:40]>> Yeah, there's x at the bottom. We don't like that. Can I move the entire set of brackets to the top? And what would its power now be? -2. So it'll be 2 4x - 3 ^ -2. Now if I write down the derivative, the power rule goes first. Bring the power down. -4 4x - 3. If I subtract one from the power, you'll get -3. Am I done? Hell no. Times the little bit more. What is the little bit more? The derivative of the inside. And what is the derivative of 4x - 3? It is four. So bang. This will give us -16 at the top.

[01:02:18]The negative says send this guy back down to the denominator. So it will be 4x - 3 cubed. All right. Are we chill with that? Great. Now I'm getting really excited cuz I can I finally told you guys like enough stuff for us to do something interesting. It's kind of like when I first teach the I don't teach English by the way. That would be the worst thing to do. I would have to give them money to sit in my class for English. But if I was to teach English, what's the first thing I'll need to do?

[01:02:47]I need to teach them the alphabet. Is that okay? Not particularly interesting, but once you know the alphabet, you can now stitch words together. You can make sentences. And that's where it gets fun.

[01:02:55]We're up to that stage now. This is like the alphabet stuff. Now we can use it.

[01:02:59]Can I get everyone's attention? First part, draw. Does anyone know how to draw the square root of 25 - x^2?

[01:03:07]Some people, some are like, I've never seen a square root having a sketch before. It's a semicircle with what radius guys?

[01:03:14]>> Five. Great. So this is simply a semicircle. So I'm going to write five five.

[01:03:22]And there's my semicircle. Okay. Next it says find a gradient of the tangent at x= 4. Can I just get everyone's attention here? Who remembers what is a tangent?

[01:03:36]It's a straight line that does what?

[01:03:39]>> That just touches the curve. Do we agree? If I look at x= 4 and I go to this point, the question is asking what is the gradient of the curve? Is that okay? At that point, I can visualize the gradient by drawing a tangent. Does that make sense? So here's the question. What is the gradient of this blue line? Which is the same as saying what is the gradient of the semicircle at that point?

[01:04:06]Does this guy help us find gradient?

[01:04:09]Not directly. What do I need to do to this guy? Differentiate it. The derivative helps us find the slope. This helps us find the what of the curve? The y-coordinate. Do you guys remember that boring stuff? I'll say it one more time.

[01:04:22]If you have y equals 2, this gives you the ycoordinate. I mean, it literally says y is equal to. If you take the first derivative, this equation is going to help you find the slope or the gradient.

[01:04:39]Because the question is asking us for the gradient of the curve. I'm not using this guy. I'm using the derivative. Does that make sense? How do I find the derivative? Do I like the look of this?

[01:04:50]No. What's what's not to like about this? The square root. Good answer, guys. Can I write it as 25 - x^2 to the power of half? Is that fair? Okay, let's differentiate. dy on dx is equal to if I bring down the if I use the power rule, I'm going to bring down the half 25 - x^2.

[01:05:11]What's the next step I need to do?

[01:05:16]Derivative of the inside. And what is the derivative of the inside? Class, be really careful here. I'll give you guys a moment to think.

[01:05:25]Yeah, derivative of 25 is nothing. It's just a number. And derivative of -x^2 is -2x. So I'm going to write -2x. Okay, watch this guys. Do you notice the half and the two will cancel? Okay, half * 2 is 1. The negativex, does that go to the top or the bottom of the fraction guys?

[01:05:45]The top. Okay, so that negative x Oh, that's a horrible second. [snorts] The negative x is going to go at the top. And where would this go, guys? With the negative power to the bottom. And it's a square root because it's to the power of half.

[01:06:01]Okay. Can I get everyone's attention?

[01:06:04]Guys, I've taught like way too many students. I've seen every problem under the sun. A lot of students think that that is the answer. It's not. This is the gradient function. But if I want the gradient, I need to sub in a value. Do you guys understand? What value am I going to sub in today?

[01:06:21]>> Four. Cuz the question says, what is the gradient at x= 4? This is the way to write it. If you want to look really cool, okay, I'll do it in blue.

[01:06:31]dy on dx when x is a value of four. This is how you can write it. So this means what is the value of dy dx when x= 4? So we're going to sub it in -4 /<unk> 25 - 4^ 2. I think that is -4/3.

[01:06:50]Is that okay? And that would be the final answer.

[01:06:54]So you need to be smart enough to realize that the gradient is not the original function, it's the [clears throat] derivative. Once you use the chain rule to find a derivative, I want you to sub in x= 4 to find the actual gradient.

[01:07:10]Yes, >> I just say yals. Is that the same?

[01:07:14]>> Oh, did it say y 4? Oh, I copied down the question wrong. I wrote x= 4. So if it was y= 4, then we'll need to sub in y= 4 to find out what x is and then sub in x. So always sub in x. I just copy down the question. So if the question was x= 4, this is how you do it. If it says y is equal to 4, then you need to figure out what the x value is and then you sub it in. Yeah. Okay.

[01:07:38]Any questions about that?

[01:07:40]Are we sure?

[01:07:43]Okay, I'll leave it for now. Next question.

[01:07:48]Let P be the point at Xals 2. Did I copy it down correctly this time? Is actually X= 2. Fantastic. Find a gradient of the normal. What the hell is that? I'll worry about that later. But ultimately guys, what's the question asking us?

[01:08:02]Find the what? Gradients. Does this give us the gradient directly? No, it doesn't. What do I need to do to it?

[01:08:10]Differentiate it. I need to get the first derivative. Okay. y is = 3 over 5 - 4x cubed. Do I can I differentiate it in its current state? No, I can't. I'm going to bring it up so we can write it as y = 3 5 - 4x ^ - 3. The next thing I'm going to do is get the derivative.

[01:08:32]Okay, guys, listening carefully. What is the derivative over here? What rule do I use?

[01:08:38]It's the chain rule which involves the power rule plus a little bit more. If I bring down a power, it will be - 9 5 - 4x ^ of -4 times the derivative of the inside, which is simply -4. Oh, why did I put a x there? -4 -9 and4 is going to give us 36. And that's going to be sent to the denominator. So, we're going to get 36 / 5 - 4x to the^ 4. Are we okay? What number do you think I'm going to have to sub in? Sub in what, guys? x= 2. If I sub in x= 2, watch what happens.

[01:09:17]If I sub in x= 2, I'm going to get 36 / 5 - 8 is -3. What's -3 to power 4? 81 or something. Does that simplify? Can anyone put that into the calculator just to make sure I didn't cook it? Oh, wait, wait. Yeah. Yeah. Yeah. That looks right.

[01:09:36]Okay. Can someone just sub in x= 2 into this function to make sure we got the right fraction?

[01:09:42]Surely >> hate doing this.

[01:09:48]>> Come on, bro. No pressure. No pressure.

[01:09:51]Did we get 36 over 81 or or did it get simplified further?

[01:09:55]>> Four over 9. Okay, great. So, it will be 4 over 9. Okay, now watch this. All right, listen. Listen, everybody. Souls back into our bodies. Eyes over here.

[01:10:05]Listen, that is not the answer cuz that's the gradient of the tangent. Yes.

[01:10:12]What's the question asked?

[01:10:14]>> Okay. What the hell is a normal? Can I just get everyone's attention? Watch this. Every time we draw a tangent line, I'm just going to use this diagram as an example. The normal is a line that is exactly perpendicular to it. That is the normal line.

[01:10:28]Are we okay with that? because they are perpendicular from like linear geometry or coordinate geo in U10. What do we know?

[01:10:38]>> Yeah, the two gradients multiply to give you negative one. So, I'm going to write it like this. Please listen. The gradient of the tangent times the gradient of the normal is minus one. Why is that always true? Because they are perpendicular lines. If I make the gradient of the normal the subject, it's -1 over the gradient of the tangent.

[01:10:58]Yes, there are two English words to help you memorize this. Number one, it is negative.

[01:11:07]Can anyone guess what's the second word I'm about to say?

[01:11:10]>> Reciprocal. Oh, very smart. So therefore, when you have the gradient of the tangent, I simply need to take the negative reciprocal. Yes. So negative, what's the reciprocal of this fraction?

[01:11:23]The flip. So it'll be instead of four divided by 9, what would it now be? 9 over4. Yeah. So therefore we can say the gradient of the normal is 9 over4. That is the answer. So the tangent you can find directly using the first derivative. If you want the normal take the negative reciprocal of that number read the question. Don't be like me where I read the question wrong. Would have lost the mark in exam. Rest in peace. Okay.

[01:11:49]All right. We've basically got one more idea I need to share and then we're we're going to wrap it up. All right.

[01:11:54]We're nearly there. If you want to take a photo or whatever, feel free.

[01:11:58]Otherwise, I'm going to rub this all out, okay? So, I can teach you the final two rules. They are basically brother and sister of each other. So, I can teach it together in one go. Okay?

[01:12:09]Okay.

[01:12:10]All right, guys. Are we ready for the last little bit of this class? Okay.

[01:12:14]Last 15 minutes. I know it's been a long class and there's a lot you had to like memorize. There's a lot of new content.

[01:12:19]Usually, we don't teach as fast. So I give you a bit more time to try questions and explore it cuz I only get two days with you and I want to teach you a lot of calculus. It's a little bit faster than usual. Okay. So does anyone know what the word product means in maths?

[01:12:34]>> It means multiply. So when we see product rule, I want you to remember it is multiplication rule. Is that okay?

[01:12:41]What does quotient mean? I mean it's like have a fun guess. It means division. Okay. So quotient rule really means to divide. I'm going to explain what we're talking about here. If I have a function where it is for I'm going to give you a basic example. Let's say it's x to the^ of 3 and x + 1 ^ 4. Is that okay? Do you notice this is two different functions of x multiplying each other. On one side you've got x cubed. On the other side you have x + 1 to the^ 4. They're multiplying. So what rule do you reckon we're going to use in this instance?

[01:13:14]>> The product rule. Now I want to say like this. You should only use the product rule if you can't easily expand it. Let me give you an example. This would be the world's dumbest thing to do. Okay?

[01:13:24]Imagine I gave you an exam and I wrote x times Oh my god, I can't write x. Uh imagine I said x * x + one and you're oh it's the product of two x's. Uh let me do product rule. I like what the hell are you doing? Because what could you literally just do to this? You bloody expand it, right? You could just write it as x^2 + x and then you can differentiate it. It will be 2x + 1. What are we even doing if you're not going to do that? Do you understand?

[01:13:52]So, so to be clear, product means multiply, but you should only use the product rule if you can't easily expand it. This is a pain in the ass cuz it's to the power four. I can't easily expand it. Let's use the product rule. Okay.

[01:14:07]How does the product rule work? Can I get everyone's attention?

[01:14:11]I'll write down a question over here.

[01:14:13]The function on the left, you're going to give it a name. Let's call it the letter U. So, this is going to be the letter U to represent the first function of X. And then we're going to use the letter V for the second function. Now, you might ask, could I use A and B?

[01:14:28]Yeah, but you'll be a weirdo. But you could. Okay. Uh you could do A and B.

[01:14:31]I've never really seen anyone do that.

[01:14:33]Hey, but you could rage bait your teacher. Uh [snorts] it would work. But let's do U and V for today. Now, here's what you're going to do. You're going to set up the formula first and then I'll show you what's going on. You're going to write u equals what is u equal today?

[01:14:45]x cub. You're going to write v equals x + 1 to the^ 4. Are we chill? You're going to write u dash. U dash. Whenever we did dash, what did it represent today?

[01:14:55]>> The derivative. What is the derivative of x cubed?

[01:14:59]>> 3x^2. What is v dash?

[01:15:01]>> What rule do I need to use?

[01:15:04]4x >> the chain rule it will be 4x + 1 to the^ of >> times the derivative of the inside but what is derivative of the inside so whatevers yeah once you've written this down you're going to write the formula so you're going to write fdash of x u-v + v d- u that's the formula so you can see the reason I make you guys do this setup is so that you can sub it in very easily is that chill what is ud dash today.

[01:15:35]Can anyone spot which one is u dash?

[01:15:38]>> So it will be 3x^2. I like to put it in brackets. v would be x + 1 ^ 4 plus and what would v dash be guys? I don't like that symbol.

[01:15:49]4 lots of x + 1 cubed x cubed. Are we okay with that? Now listen, that is the answer. But I hate it. Why?

[01:16:01]You'll learn tomorrow in the world of calculus that we're going to need to find more stuff than just the derivative. And so presenting your answer like this makes your life really, really difficult. Is that okay? Here's my suggestion. After you sub it into the product rule formula, I want you to factoriize. Factoriize means look at each of these terms and ask yourself, what do they both have in common? Can you guys spot one thing they have in common? What is it? x squ. There's an x squ. There's an x cub. They both have x squ. Great. Whatever they have in common, I want you to take it out. So, take out the x squ. What else do they both have in common, guys? x + one to the power of three. Once you've taken out the common factors, I want you to use a square bracket.

[01:16:45]Can anyone tell me what is remaining on the left term? If I took out the x squ and three lots of the x+ one, >> it will be 3 of x + one. Is that okay?

[01:16:55]The three is going to remain. If this is x + one to the^ 4 and I've only taken out three of them, there's one left. Did you guys follow that? Over here, if I've taken out the x squ, what's going to be left, guys? Four lots of what? X. Is that okay? Finally, I want you to expand and simplify. So, that'll be x^2 x + 1 cubed 3x + 3 + 4x is 7 x + 3. And that is a much nicer expression. I mean, look at that. Doesn't that look a lot simpler than whatever the crap this is? Yes.

[01:17:29]Okay. I'm now going to break down what I did into individual formulaic steps that you can follow every single time. Step one. What do you reckon guys is a good name for this first step?

[01:17:43]>> Set up.

[01:17:44]>> Great. I was going to say the word setup, right? Setting up the question means write down what you need. You know, when we did first principles earlier today, we did f ofx and f of x plus h. That's the same thing. It's setting up the question. If you do set up, you make less mistakes. Make sense?

[01:17:58]Okay. So, the first step is set up your product rule ingredients. Number two, what is this called? What do you reckon is a good name for this one? The bloody formula. Okay, we love that. And then what's the next step? Substitution.

[01:18:13]These are the exact same steps we did for first principles. By the way, you can look back at the working out. You'll realize it's exactly the same. And then after substitution, there's a new step.

[01:18:21]What is this? I'm going to write it in bold capitals to scream at you guys.

[01:18:25]What did we do over here?

[01:18:27]We factorized.

[01:18:31]And then after you factoriize, the last step is simply Oh, what? Simply to simplify. Lovely English. Okay. Is to simplify. Do you guys understand? So set up the question, write down a formula, sub in everything, do factorization, simplify. Okay, great.

[01:18:51]That's the five-step process for product rule. Any questions about that, guys?

[01:18:56]Sweet. Can I get everyone to try question? Oh, example 11. Just do one for me and then we're going to move on.

[01:19:05]Okay. Example 11 just cuz I'm running a bit shy on time.

[01:19:13]As I walk around, you guys better not do me dirty. I want to see you guys do all of these five steps. All right. Did we go okay with that one? So, you've got your u and your v and then you're subbing it back into the formula and factorizing and stuff like that. Okay.

[01:19:29]Does anyone want to just ask any question? Any any questions? Okay.

[01:19:35]All right. All right.

[01:19:38]It's okay off camera. If you want to ask a question, you can ask a question if you're too shy. Okay. I've got one more formula I need to teach you. Holy heck.

[01:19:45]This has been freaking hectic. But that's the product rule. We're down to the last one. Quotient rule. Can I get everyone's attention looking up here and I'll show you the formula.

[01:19:55]Quotient rule is used when we're taking the when we're taking the um division of two functions. The function at the top is called u and the function at the bottom is called v. So if I look at this example over here, can anyone tell me what would u be? Well, I just bracketed it. And then v would be x - 3. Is that okay? So function divided by another function. The top is u, the bottom is v.

[01:20:20]Do you guys notice that the formula is just the same as the products rule except there's a negative instead of a positive. Do you guys see the product rule is u-v plus v-? What is the formula over here? U-v minus v d- u. So I want you to recognize something. This is the same as the product rule.

[01:20:45]but with a negative symbol. Are we okay with that? And the second thing is the entire thing is divided by v^ squ. Do not forget that. All right, good. Now, because of timing, I'm not actually going to do this question. I'll do this one up here. So, what would u equal to, guys? And what would v equal to? Let's write it up. So, set up. What is u?

[01:21:07]x. And v would be 5 - 3x. What is u dash? and v dash would be carefully negative3. Is that okay? So once you've done your setup, we can say dy over dx is u-v do I do plus or minus? Minus for the quotient rule. v-u all over v ^2.

[01:21:29]What is the next step after we write the formula class? We're going to do substitution. Let's sub in everything from the setup into the formula. U dash is 1 5 - 3x minus - -3 and then x. Okay. All divided by v ^ 2 which is 5 - 3x^ 2.

[01:21:49]Okay guys once I've done substitution what I generally want to do is I want to factoriize or simplify. Now at the top over here you'll notice that there's just not they they don't have anything in common. So therefore you're going to skip to the next step which is expand and simplify. So this over here is going to give me 5 - 3x. What's - 3, guys? + 3 * x would be + 3x all over 5 - 3x^ 2. Is that okay? What do you notice between a - 3x and a + 3x? So you're left with 5 / 5 - 3x all^ 2. Okay. And that's what the question actually said. Show that the derivative is that expression. And that's exactly what they wanted. Okay.

[01:22:33]Last bit. Find a gradient. Guys, for the gradient, do I use You've got two options. Y dy dx. For the gradient, which one am I going to use?

[01:22:48]For the gradient, which one am I going to use, class? dy dx. Fantastic. Okay, good. We're going to use dy dx, which we already found. Is that okay? And we want to find the tangent. Tangent, I can use this guy directly. Is that fair enough?

[01:23:01]So I'm going to write it like this. What is the value of dydx when x= what value?

[01:23:07]The point is k of 2 -2. So what value of x are we going to use class? We're going to use two. Great answer. If I have two, I'm going to sub it into here. That will give us 5 over 5 - 3 * 2.

[01:23:23]So it'll be that. This will give us what? Five. Okay, great. This is the gradient of the tangent or the normal.

[01:23:30]Which one?

[01:23:32]>> Tangent. If I want the normal, there were two words. Do you guys remember the two words?

[01:23:37]>> Negative reciprocal. My man. So if I want the normal, I'm simply going to take the negative reciprocal of this and I'll get the answer to the second part of the question. So we can say like this. Number one, the gradient of the tangent is simply five. And number two, what is the gradient of the normal guys?

[01:23:54]>> -1 over 5. And boom. That is the end of day one calculus. Okay, good. Write it down. Copy what you need and we'll wrap it up here.