[MUSIC] Equations and functions are clearly related to change in one variable when another variable or other variables change. This week dealt with this concept in detail by introducing the concept of a derivative. This is a crucial topic in many scientific disciplines, including finance, business, and economics. We want to know either how much the variable costs change if the inputs change or how much revenue changes if production level changes or how much consumption changes if income changes. Or how quickly some or even all of these quantities change over time. For computing the future position of a planet in space, or to predict the growth in population of a species, or to calculate how the demand for a commodity changes if the population changes, or finally, how much capital we will have to repay because of interest payment. We need information about rates of change, all variables and functions. So this week, we have defined the concept of the derivative as the rate of change of a function. The week has been focused on the basic definition of the derivative, and we have been giving the geometrical interpretation for this concept. This representation is very intuitive. And it gives insights on why. When we started the graph of a function, we would like to have a precise measure of its steepness at a given point in the graph. From this graphical intuition of a derivative, we have built the idea of incremental ratios. And have been seeing the derivative can be seen as the limit of this ratio in a given point of the function. From this very general definition of a derivative, we have presented some important rules for calculating derivatives. And we have shown how to compute the derivatives of a function. In more detail, we went through rules of differentiation, such as the product rule and the quotient rule and the chain and power rules. We also focused on the idea of calculating the derivative to measure the rate of change of a variable or a function. Finally, we introduced the concept of higher order derivatives and presented all of these concepts using some examples. These examples have been taken from business, economics, and finance. Through these examples, we saw in detail how to interpret the derivative and why it became a mathematical tool of crucial importance to the development of modern modelling for finance and economics. Up to this point, the material presented has dealt with mathematical models but have a relatively low number of variables and functions. The next week, we'll present a tool that allows a generalization of the concepts we have been presenting so far. Thank you very much. [MUSIC]