Before we begin, I'm going to recall some things we're going to need and what is to follow. A couple of things to remember. The limit as h goes to zero of Sine of h over h. At the time we did it by graphing, we looked at the graph. This function does something like this, whereas a hole in the graph at zero, you can plot this thing and look at it. The hole on the graph is zero, I can't plug in at zero, division by zero is a problem. But if you look at the graph, it really, really wants to go to one in the limit. So in the limit, this thing is one. Keep that in mind, we're going to use that in a second. Then, I also want to recall something that, kudos to you if you remember this, from pre-calc, from trig, I don't know, wherever you saw this. There's an expression, it's good reminder anyway. If I add inside a trig function, this is a, I think they call it the sum formula or something like that for trig, that the key is hopefully your pre-calc or trig teacher yelled you enough, you can't just bringing this in, it's not as simple. Sine and Cosine don't behave that nicely. The addition formula. When you do this, it becomes Sine of the first times Cosine of the second. This has got to be on that trigger view sheet somewhere, but you can find it or remember it, plus Cosine of the first, and then times Sine of the second. So these are the two little things that remind you what they are. Hopefully, you remember that you can't bring the Sine in, it's actually a formula to do this. So now with these two things ready to go, let's study the function, Sine of x. So we want the derivative d, dx of the function Sine of x. Once we get Sine, Cosine is very similar. So I don't know how to do this. Well, let's just plug and chug and hack at it using the limit definition. So the limit definition would say that this is equal to the limit, as h goes to zero of the function plugged in at x plus h, minus the original function, all over h. Great. What's that? I don't know. Well, you can see why I want to remind you what the formula is for Sine of x plus h. Because that trig identity, which of course, we've all remembered, is going to come in handy right now because I want to rewrite this Sine of x plus h. So remember it's Sine of the first times Cosine of the second, plus Cosine of the first term, and then Sine of the second. So you have to know a little bit of trig to be able to do anything with this limit. Then Sine of x, no change there. This is all over h. Now, I want to regroup like terms. So we want to break this up into a limit. Do they have anything in common? Well, the first and third terms have a Sine of x. So let's regroup the first and third terms, and I'll do that in one shot. I'll factor out the Sine of x. So what's left? I have Cosine h, minus 1, all over h. So I group the first and third terms. Then, that middle term, let's write that as its own thing because it doesn't have anything in common. So this becomes Cosine x, but I'm going to write it in a way where I can use the limit that I know. I'm just going to split off. Well, not split off, but I'm going to write together Sine of h over h. Now, instead of one difficult limit, I've kept two limits and they both have trig, good times. But I can actually work with this thing. Remember Cosine is on X, at least for the second term. So that doesn't play a role as h goes to zero. As h goes to zero, the limit that I care about is Sine of h over h. So that limit from our analysis or limits before that goes to one. So the expression on the right is just Cosine of x plus Cosine of x. Then as h goes to zero, once again there is not like sine of x doesn't play a role. Sine of x is not affected by the limit as h going to zero and so I need cosine of h minus 1 over h. Now this one I didn't write it down but you can graph this as well and see this limit actually wants to go to zero. It's not immediately obvious and we don't have the best tools at the moment to see that, but I promise you there will be a time we come back and say, "Oh, that's obvious." For now I would say look at the graph of this thing. This will go to zero and what you end up with is sine of x times 0 plus cosine of x times 1. Guess I didn't use that, better known as cosine of x. Stare at this for a minute because this is mildly amazing. Although maybe if I asked you to guess this you would have guessed this, but the derivative of sine is cosine. Does that surprise you? No, I don't know. That surprised me a little bit the first time I saw it, that one of the relationships between sine and cosine in addition to all the trig formulas is that sine derivative is cosine. Add that to the list of derivatives that we know. For similar reasons, I'm not going to go through the whole calculation but for similar reasons, I want to give you one more. The derivative of cosine is, what do you think it is? Most of you will say sine. It's actually negative sine. It's negative sine of x. Why is it negative? In the sum and addition formula for cosine, there's a minus sine in there. If you go and actually work this out, if you have x plus y, it's like thing minus the second term. That there is a plus for the sine, there's a minus for the cosine formula. That minus follows its way through the calculations and leads to a minus sine of x. This is the derivative of sine of cosine. Derivative of cosine is minus sine. The derivative of sine is cosine. Once you know those two things, with all the other stuff you have, you can now find like lots of cool stuff. For example, what's the derivative of the tangent function? You say, "Oh man, we got to do the whole limit definition." No, don't do that. Tangent is sine of x over cosine of x. Well, it's a quotient. How do you do derivative quotient? Quotient rule. It's the bottom function times the derivative of the top sine minus, be careful, the top function sine of x times the derivative of the bottom. The derivative of cosine all over the bottom function squared. I know what the derivative of sine is. This becomes cosine times cosine. Cosine x times cosine of x minus sine of x plus Theta of cosine. Remember it's minus sine over cosine squared. Clean this up a little bit. Cosine times cosine is better known as cosine squared of x. Two negatives make a positive and you get sine squared of x all over cosine squared of x. There in the numerator is cosine squared plus sine squared. That's one identity from trig that hopefully you remember if you just know one of them, and it becomes one over cosine squared of x. That is better known as secant squared. The derivative of tangent, putting all these pieces together becomes secant squared. This is like the beauty of calculus where if I know just certain pieces, I can find almost anything I want. We can do that with not just tangent, but any of the reciprocal functions. I'll just do one more and there's nice about this is if you know, let's say you had a test and the derivative, you need the derivative of tangent. You studied really hard and you know the derivative of sin and cosine, like you can recover. You can go through this calculation. This isn't too horrible to do and you can find or write down the derivative of tangent, like you'd go off on the side real quick and do this and if you forget derivative of tangent you just discovered, you just calculated it's secant squared. This is important to understand when these things come from because it reduces the dependency on memorization. If you only memorize, like what the derivative tangent is and you forget it on a test or something, then you're shot, there's no recovery. But if you understand where it comes from, my God, it's just the quotient rule. Then you can do that. Let's do one more. How about secant? Why not? There are six trig functions, but we'll just pick one here. The derivative of secant, let's say, view as a quotient. So one over cosine of x and let's do the quotient rule. It's the bottom function, cosine of x times the derivative of the top. Now that's derivative of one that's a constant just going to evaluate that, that's zero minus the top function times the derivative of the bottom. So minus the top function times the derivative of the bottom. I have it up here, so I'm not going to write it. The formula, I'll just put the derivative in there, sine of x. So bottom times derivative of the top minus top times derivative of the bottom, all over the bottom squared. Clean this up. Cosine x times zero, that goes away. Two negatives make a positive. You get sine of x over cosine squared of x and that's perfectly fine. They don't leave it like this, they actually break this off. They write it as one over cosine of x times sine of x over cosine x. So you split up the denominator and the reason for doing that is one over cosine x is secant of x and then sine over cosine is tangent of x. The derivative of secant becomes secant tangent and we can go through this for all the other trig functions. I mean, once you've seen how they are done, I leave it to you to work this out, let me just summarize what we know about trig functions and you give you this as sort of a rule book. The first ones that we put down, the derivative of sin is cosine and the derivative of cosine, be careful, is negative sine of x. Using that information, you usually go off and you find the derivative of tangent, and that's secant squared and then you do the reciprocal functions. The reciprocal of sine is cosecant CSC and that one turns out to be, no, we didn't work this out, so I leave it as an exercise, but it's done almost the exact same way. Maybe we'll do as a warm up in the next video, but it's negative cosecant x times cotangent x. So negative cosecant x times cotangent, the reciprocal of cosine is secant. We did that one and we saw that this is secant tangent and then the derivative or the reciprocal of cotangent, try again, the derivative of tangent is cotangent of x and we didn't do this, but you can check, do the quotient rule on cotangent, which is just cosine over sine minus cosecant squared x. So we'll work some of these out in future videos just for practice. But for now, I just want to get them all in one place. As you go through these again will hopefully uses so much that they become second nature, but have these handy, put them in your rule book. I just want to point out one thing that I'm embarrassed to tell you I didn't realize until a student pointed out maybe like, after the thousand time I taught this class, but the ones with a negative sign. This one, the cosine has negative sign. Cosecant has a negative cosecant cotangent and cotangent, which is negative cosecant squared. By just pure coincidence there all the trig functions that start with a C. My students like, "Is it a rule that if you are a trig function and start with the C than your derivative has to be negative?" Like that's not a rule, but it seems to work out just by coincidence. So just something to help you remember where the negatives go. If you start with a C based on this list, then your derivative has a negative sign on it. So don't forget cosine, cosecant and cotangent have a negative with their derivative. Keep those in mind, keep this handy and then we will do lots and lots of examples with them in the next video. See you there.
