[MUSIC] So, last time we dealt with derivatives. I gave you some definitions of what derivatives are and to give you a bit of content and environment as to where derivative can be important. Let's have an example. So this is average revenue function which is represented by 15 as a value and q as the quantity. Now you as the producer if for instance you are producing something would be interested to know what effect the production has on your revenue. And you have an average revenue. What does it mean? It means on an average when you increase your quantity by say, 1 unit, you decrease your average revenue by 1 unit respectively. But it is it through for all the units? Is it true that if you produce say thousands units or millions units, the effect will be the same. And this is where marginal revenue or derivatives are quite important. Let's start first of all by giving it a content that relates to the two examples that provide you earlier. Simply saying that one function of a our interest would be linear function and in other function, our interest will be quadratic function. So for instance, when we talked about linear functions Regardless where you take the points, The rate of change will be the same. So, here or here, it will be the same, and it will correspond to the gradient of the linear function, which is represented by, This letter a. In quadratic function, if you remember, If for instance, this is a quadratic function, Which has the following form, the gradient will depend very much, On your location, On the graph. So average revenue can be quite deceiving because it depends where you take this average revenue from. So to get a better idea of what effects your production has on your revenue rather than looking at averages which is basically saying what is the gradient of those lines. Or those lines which is represented simply by functions similar to that. We would like to have a look at specific points on the graph. And not a linear approximation of two points which are near each other. So, let's try to work out with this average revenue function. [COUGH] This average function as I said represents a similar function to let's say, that if we increase production by 1 unit, our average revenue will decrease by 1 unit respectively. But whether this is the case for all the output points, this is the question that we're after right now. So in order to translate this average into total revenue, what we need to do, We need to multiply this by the quantity itself. Because what is essentially average revenue? Average revenue is total revenue divided by quantity. Therefore, total revenue = average revenue times the quantity, which is exactly what I did here. [COUGH] So in this case we'll have 15q- q squared. Now this is the total revenue function. For this total revenue function, we can find out what will be the rate of change for each and every production unit. Or in other words, we call this a marginal revenue function. Sometimes marginal revenue will be the same value as an average revenue but as I said before, it's not necessarily so. So in order to find the margins revenue, what we need to do, we need to find the derivative of the total revenue function. Because now you're already in position to know what were the derivative of linear function and quadratic function, I can directly go into finding that this will be 15- 2q. Now as you see this and this function are very similar. However, there is a big distinction between those two functions is that the value that stands in front of q, in front of production is 2 here whereas as it's 1 here. And what does this function say which this function doesn't? This function can say to you what will be the effect of production unit on your revenue, regardless where you take the production unit from. Whether you take the production unit from 100 to 101 unit, or whether you take the production unit from millions and you increase it by a tiny bit? So this will be a much more accurate representation of the fact on your revenue on the rate of change in revenue, compared to this one.