So this segment we're going to look at Calculus. Calculus is an area of mathematics that allows you to analyze change. Really, it's the study of how things change. One of the main components of calculus at least as we're going to be examining it, is something called a derivative or engaging in differentiation. One of the main purposes for taking a derivative is to essentially identify how something is optimized. So if we're looking at businesses, we might ask the questions, where does the company profit maximize? So that's an optimization. Where do they cost minimize? How do they maximize shareholder value? So there's a number of different reasons why we would engage in the practice of calculus in a business setting. But learning calculus is not too difficult. You probably will use it a handful of times during an MBA course of study or a study in business school and some other degree. So let's look at some of the very simple rules with respect to calculus, and then we'll go into the rules a little bit more in depth. So the first thing we're going to look at is the rate of change and derivatives. Now, one of the things that we have to be comfortable with is understanding the language of mathematics and what we're talking about. So here, we say that Y is a function of X. So I've written this thing right here, Y is a function of X. Now, this is the mathematic way of saying that there's some relationship between an X, some variable X and some variable Y, and we call that relationship a function. So we could describe the relationship between X and Y just by writing literally Y is equal to f of X. Y is a function of X. Sometimes, you might have a more explicit relationship. For example, Y is equal to three X squared down here. That is a relationship between X and Y. What this means here, I'll show you an example is, let's suppose we have some X values, and we then apply the function from those X values. So we have 3X to the power of two, and we get some corresponding Y value. So we say, look, let's suppose that X is equal to one. So we take 1 to the power of 2. So one times one is equal to one, and multiply that by 3 and so then Y would be equal to three. So the relationship 3x squared means that if I put an X of one into this and I do the multiplication. I do the X to the power of two, multiply by three, Y becomes three. If X were two, then I would say two to the power of two. So 2 times 2 is 4 and then 3 times 4 is 12. So then Y would be equal to 12. So I can plug in any value of X and I get a corresponding value of Y by taking X through this multiplication process, raising it to the power of two, multiplying it by three. So when I say that Y is a function of X and I give you the specifics of that function, I'm really identifying a relationship between one variable X and another variable. Y. Here we have a change of X from X of one to X of two. Here we have change of Y from three to 12. What's happening here is that every time X changes, Y also changes. So what we need to know is we can calculate the derivative of Y with respect to X, dy, dx. Here's my equation. Y is equal to three X squared. Now, when I take the derivative, I am going to engage what is called the power rule. In this case, the power rule says this. Let us suppose that Y is equal to some function of X, where K is a constant, X is my variable and N is the power. So in this example, this three right here is the K and that two is the N. When I take the derivative of Y with respect to X, what I'm going do is I'm going to drop this power two in front of my coefficient. So this guy drops down here and then I multiply those two together. Then what I do is I subtract one from my power, minus one. So now I've got this dy, dx is equal to 2 times 3 times X to the power of 2 minus 1. So that's A, becomes one. So the derivative of Y with respect to X equals 6 times X. So my dy, dx equals 6X. How does my Y change every time I increase an X? It changes by six times that value of X. That's how I calculate a derivative with respect to the power rule. Now let's look at why we would do doing this, and then let's look at a few examples.
