 Welcome back to our lecture series, Math 12-10, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. This video starts lecture 17, which is the second part of a special three-part lecture, introducing the notion of a derivative. In our second chapter, we've been focusing on the idea of limits, and so now we're ready to talk about the idea of a derivative, which is going to be a limit of a difference quotient. So what I want to do before we define what a derivative is properly, I want to remind ourselves about some of the notation that's going to be relevant to discussing derivatives, and also give you some idea why we are interested in the first place. So imagine that we have two points, x1, y1, x2, y2, and imagine these are points on some function maybe say f, like so. In which case, we might have the point x1, y1 over here, and we have some other point right here, let's call it x2, y2, like so. So if we measure how far apart these two points are just along the horizontal, so think of the x-axis as being somewhere below these things. So maybe the x-axis is just down here, right here, and so let's measure just the horizontal change. So we have here x1 and we'll have here x2, and so the distance between x1 and x2, this would be, well, just take the difference of the two things, the distances, the difference here, x2 minus x1, this is what we're going to call delta x, right here. That triangle symbol is the Greek letter delta, which here is meant as a mnemonic device. Delta, da, da, da, delta is supposed to be like the Greek equivalent of the Roman letter D, and so D is for difference in the situation. Similarly, if we consider the difference of these points along the y-axis, we might get, let's put our y-axis in there, something like the following. We get right here, and so let's say that the y-coordinates, we have a y1 right here, we have a y2 right here, and so the distance along the y-axis, we'll call that delta y. Again, that's the difference there, y2 minus y1, and so we're just measuring how far apart these things are. Now, if we want to compute the so-called average rate of change, this is given by the notation delta y over delta x, and this is because we take actually delta y and we divide it by the quantity delta x, so this will look like y2 minus y1 divided by x2 minus x1, which the significance of this geometrically is that we were to connect the dots from x1 and y1 to x2, y2. We get this so-called secant line that we've talked about before, and the slope of this secant line, it's slope m is gonna be delta y over delta x. The average rate of change measures the slope of the line going from one point to the other point. Essentially, if we just kind of flattened the function, on average, if we view it as the same growth as a line, then the slope, the growth of that line would just be this average rate of change. Let's take a slightly different perspective now. So instead of focusing on this average rate of change, let's transition our perspective to the so-called instantaneous rate of change. What if we only have one point in consideration here? What if we only care about the point x1 comma y1? And we're interested in this, say, the so-called tangent line that'll exist at x1, y1. How would we figure out what that would be, this instantaneous rate of change, which would be the slope of the tangent line there? Well, imagine we take points on f, y equals f of x, but we allow them to get closer and closer and closer to our point x1, y1. This would have a consequence on the x-axis, we're allowing our x-coordinates to get closer and closer and closer to x1. Essentially, we're gonna be looking at the limit as our second point, x2, converges here to x1. So instead of focusing on x2 itself, because x2 is not the point we really care about, let's focus instead and say that x2 is equal to x1 plus h, which h here is just measuring the distance between the fixed point, we have this fixed point right here, that is to say x1's not gonna move, but x2 is allowed to move. So we see that x2 minus x1 is just equal to h. So h is basically just serving the same role as delta x right here. Why do we use a second symbol to represent the same thing? Well, it's common notation in a calculus one setting right here. So h is just measuring the distance between x1 and x2. So as we allow h to go to zero, this would imply that x2 is approaching x1. The two statements are one in the same statement right here. h goes to zero, delta x goes to zero, implies that x2 approaches x1. And so then you're gonna see that as h gets smaller and smaller and smaller, these points are getting closer and closer and closer to x1 right there. And so if we take a point that relatively close here and we connect the tangent, the secant line there, as x2 gets closer and closer to x1, as h gets smaller and smaller and smaller, this secant line will better and better approximate the tangent line that we'd seen before. So in fact, if we take the average rate of change, delta y over delta x, as x1 ranges from x2, this will look just like the slope form that we're calculating the slope of the secant. But since the y coordinates come from the function y equals f of x, y2 is just f of x2 and y1 is just f of x1. And so then if we make the substitution that x2 is just x1 plus h, it's just a little bit to the right of x1, then we can make this substitution right here, x2 will become x1 plus h, right? In the denominator, of course, you'll see x1's cancel. So we left with just an h and then we get this difference quotient right here. So it's important to realize that these two difference quotients, the slope formula and what we've usually been calling the difference quotient in this lecture series, these are one and the same thing. The h just represents a small distance between the first point and the second point. And so this second perspective is extremely useful as we try to compute the instantaneous rate of change and hence the tangent slope. And so this will be denoted dy over dx. So we often will write for the average rate of change or the instantaneous rate of change. We put a little subscript here to denote which rate of change are we talking about? So delta y over delta x, as we go from x1 to x2, we need to mention that because the average rate of change depends on the interval. If the interval is clear from context, we might drop it, but we should specify it. The same is also true for the instantaneous rate of change. We'll use a dy over dx. The d here is still representing a difference, the distance of the y coordinates divided by the distance between the x coordinates. But we do use a lowercase d when we're trying to describe the instantaneous rate of change and we use the capital delta for the average rate. But we might need to mention for the instantaneous rate of change, where are we measuring the instantaneous rate of change at? Well, here we'll say when x equals x1, because x1 is a specific number. And here it's not a variable, we just don't know what it is. So I'm using a symbol x1, x2 to talk about these numbers. And so this would normally be like, oh, what's dy dx when x equals two? That's normally what it would look like for us here. And so what is this instantaneous rate of change? It's the limit of the secant slopes. That is the tangent line will be the limit of the secant lines. And so the tangent slope will be the limit of the secant slopes. That is the instantaneous rate of change is the limit of the average rate of change as the second point slides closer and closer to the first point. So that as we slap this limit in front of the average rate of change, in summary, we're just saying we're taking the limit as here delta x that goes to zero of delta y over delta x. That's what the instantaneous rate of change is. That's what dy over dx is. We take the limit as delta x goes to zero right here. If we use the original slope formula, we get something like this or if we use our modified difference quotient, we get this one right here. The second form is actually gonna be the more useful approach for us here. Because again, when it comes to simplifying these difference quotients, we're gonna prefer this one because this difference quotient, if I just plug in h equals zero, you're gonna end up with a zero over zero. Consider that in your mind how that happens. And so it's important that we don't just plug in h equals zero. We have to algebraically simplify the difference quotient first, like we've seen in previous exercises. I often mention how pre-calculus is essentially like the karate kid that we are just practicing karate out of context. Now we're in context. Now we're ready to fight the Cobra Kai here because we can use those algebraic simplifications, those algebraic tools to help us start simplifying and computing limits, such as a derivative. So let's define it properly then. What is the derivative? So given a function f, we define the derivative of the function f at the point x to be the limit as h approaches zero of f of x plus h minus f of x over h, which this limit of a difference quotient should hopefully look familiar. This of course right here is just dy d over dx. This is the instantaneous rate of change at x. And so that's what the derivative is. The derivative is the instantaneous rate of change of the function at a specific point. So again, x will be a specific point, but we're just using the variable because we don't know what that point is at the moment. So we take the limit of the difference quotient, like we saw in the previous slide, the limit of the difference quotient is gonna be the limit of the secant slope. That's what the tangent slope is. So the derivative measures the slope of the tangent line. And we often abbreviate this as f prime of x. So when you see f prime of x, that means the derivative of f at the number x. Now it's often the case that the limit of a function does not exist because the derivative is defined upon a limit. If that limit doesn't exist, and we've seen examples where that happens, if the limit doesn't exist, then the derivative doesn't exist as well. So we might say that a function is not differentiable. So again, that's a terminology we should use right here. If the derivative f prime of x exists at the number x for our function f, we say that it's differentiable. f is differentiable at x if its derivative exists at that point. If it's not differentiable, I'm gonna say it's non-differentiable at that point. And we'll see some examples in the next lecture of why a function might not be differentiable at a point or not. The process of computing the derivative f prime is called differentiation. And the branch of calculus that revolves around the notion of the derivative is called differential calculus. It's important to note at this moment that the derivative is in fact a function of x. Since f prime of x varies when x varies, this differs from both the slope of the tangent line and the instantaneous rate of change as we described them earlier, either of which is represented by the number f prime of a. So if a is a specific number in the domain of f, then f prime of a, this is the slope. So this is the slope of the tangent of this tangent line at x equals a. So the specific number is the slope of the tangent line. f prime of a, this specific number is the instantaneous rate of change of the function f at the number a inside the domain. So the slope of a tangent line, instantaneous rate of change, these are specific numbers. f prime of x on the other hand is the formula that gives us the arbitrary slope of the tangent line or the arbitrary instantaneous rate of change. So the derivative is the function that gives us the slope of the tangent line. So with this perspective, the derivative has two very important interpretations. First, the function f prime of x represents the instantaneous rate of change of y equals f of x with respect to x. This instantaneous rate of change could be interpreted as a marginal cost. You know, if the original function f is a cost function or it could be the velocity function if the original function is interpreted as a position function or a displacement function. Now, second, the function f prime of x represents the slope of the graph, f of x at any point x. If the derivative's evaluated at the point x equals a, then f prime of a, as we were talking about earlier, is the slope of the tangent line of f at x equals a. In particular, the slope of the tangent line of f at x equals a would have the following formula. So our tangent line, it'll look like y minus f of a equals f prime of a, whoops, f prime of a times x minus a. So this is the tangent line at the point a comma f of a. So because we have a function f, the y-coordinate given the x coordinate a will be f of a. So with our usual slope point form, we have y minus y one is equal to the slope times x minus x one. X one, y one are a specific point on the line and m is the slope, which in this case is given by the derivative. I should also mention that there are several commonly used notations for derivatives of a function f. We've, we're often gonna use the notation f prime of x. There's a huge convenience there. So that's one of the most popular notations. Another popular notation we've seen already is dy over dx. This kind of resembles the, this resembles the average rate of change for which we derive the derivative from the instantanet rate change. Sometimes you see things like df of x over dx because y is f of x, but sometimes we wanna be specific about the function. And with this perspective, we also sometimes see d over dx of f of x, something like this. So you're kind of treating the derivative as this like operator. The derivative is a process that changes a function f to a new function f prime, it's derivative. Other notations include things like d sub x of f of x or d sub x of y. D, the capital D here represents the derivative of course. The subscript here represents, we're taking the derivative with respect to the variable x. This is something that'll become more clear in the future when I say we're taking the derivative with respect to a specific variable. This will be particularly important in the context of multi-variable calculus. So if there's more than one variable, we might be interested in the rate of change of one variable, but not necessarily the rate of change of a different variable. So we sometimes need to be specific on what's the variable we're taking derivative with respect to x in this context, all right? And so this is overtaking the derivative with respect to x. If we're taking, let's say we have a function y equals f of t, f of t, excuse me, like maybe this is some type of like position function, maybe it's like the variables are s equals f of t. Take like a physics example, this is some distance function with respect to time. In that situation, the derivative s, we would say like s prime, this is f prime of t, we can say something like that. We could take the derivative ds dt like so, we could take the derivative dt of f of t, something like that. So we might change the variables, but the notation would change appropriately and that's perfectly fine with things like this. I should also mention that because the derivative itself is a function, it makes sense to take its derivative. We could take the derivative of the derivative, which is the so-called second derivative. The second derivative, this here is going to be what we would commonly call f double prime of x, this is going to be the limit as h approaches zero of f prime of x plus h minus f of x, sorry, f prime of x over h. That is to say, this is the derivative of the derivative. So you take f prime of x and you take its derivative because as the function, the derivative is a function, it could have a derivative as well, so we could take that derivative. Using some of the other notations we had, the second derivative could look like d squared y over dx squared, that'd be the second derivative, or you take like dxx of f of x, some of the alternative notations here, but you know what, since the second derivative is also a function, because it's the derivative of the function, we could talk about the third derivative, right? The third derivative, in which case, you would probably write this as something like f triple prime of x, in which case this is going to be, take the second derivative, you take its derivative, or we could say like d cubed y over dx cubed. This pattern continues on and on and on. We could talk about the fourth derivative, the fifth derivative, the sixth derivative, we could continue on and on and on. When you start taking higher derivatives, this little tick mark she put here for the primes, they kind of look like Roman numerals, so like with the fourth derivative, someone might write like f one four, so kind of like Roman numerals, but it's more common to write it as with parentheses, f parentheses four of x, and specifically you're talking like the nth derivative, you'll get f to the n right there of x. So this would be the, you take the derivative in times to the function. Now why in the world, what you want to talk about the second derivative, the third derivative, the fourth derivative, and I know it seems a little premature at this moment, because at the moment, we barely just learned what a derivative is. To compare it to other derivatives, the derivative is sometimes called the first derivative of the function. So take the following example from physics. Earlier we saw that the first derivative, that is the derivatives we just defined, the instantaneous rate of change of a position function, the first derivative would give the velocity of a particle. So for particles moving, you have s equals f of t, this is gonna be the motion, the position of the particle. We've seen that it's first derivative s prime, f prime of t, this gives us the velocity of that function. It tells you how that function's position is changing with respect to time. So we might call that v. But then if we take the derivative of velocity, velocity prime, which would be the second derivative of position, this is what we call acceleration. Acceleration is measuring how the velocity is changing with respect to time. So for example, if velocity is positive and the acceleration is positive, that would imply that the velocity is increasing so that the object is speeding up. If the velocity is positive and the acceleration on the other hand was negative, because these again are vector quantities, they have direction to it. If your velocity is positive, but your acceleration was negative, that actually means that your object would be slowing down. In particular, when velocity and acceleration point in the same direction, your object is speeding up. That includes whether they're both pointing forward or they're both pointing back. But on the other hand, if velocity and acceleration are pointing in different directions, that means the object is slowing down. But why stop there? If you could talk about the change of acceleration with respect to time, that would be the second derivative of velocity. That would be the third derivative of position. This is what we commonly refer to as jerk. So like if you like amusement parks, like Disneyland or something, that jerk you feel on the roller coaster, right? That sudden change of acceleration, that's the thrill we get when we're in a high speed car chase or we're on a roller coaster or other amusement park rides. It's that change of acceleration that I'll commonly refer to as jerk. It's never a good thing to call someone, you know, the third derivative of position. It's a rude thing to say. But the point is these higher derivatives, second derivative, third derivatives, fourth derivatives, et cetera, they do have physical and practical applications. Now for the rest of this chapter, chapter two, we're gonna focus just on the first derivative because we don't yet know how to calculate first derivatives very effectively. But what we'll see over the end of this chapter and the next chapter, number three, we're gonna learn how to calculate derivatives more effectively, we'll get better at it so that then it doesn't become too prohibitive to start calculating things like second derivative, third derivatives, fourth derivative, et cetera. So let's take a look at some more videos in this lecture here to learn how to calculate derivatives.