 Welcome to the fourth session of managerial economics. Basically, we are on the first model of managerial economics which talks about introduction and fundamental to managerial economics. So, if you remember in the last class, we just discussed about the functional relationship between the economic variables, how they are related, what are the different forms to represent them and then we discussed some of the important economic function like demand function, bivariate demand function and multivariate demand function. So, today's session will focus on the different type of function gets used in a typically in a demand function. How to measure a slope and what is its use in the economic analysis? Different method to analyze the slope, find out the slope or measurement of the slope. Then derivative of the various function and in the next session will basically take the optimization technique and constant optimization. So, till now all our discussion if you look at it just focuses the demand function, but apart from the demand function in there are certain other topics also where we generally use the relationship between two variable in a functional form like production function which represents the relationship between the inputs like labor and capital with the output. We talk about the cost function where it is basically the relationship between the output and the cost of the production associated with that. Then we talk about the total revenue function and its represent the combined function of quantity, produce and price function based on the demand function. And sometime also we talk about a profit function and this is the profit is basically as you know it is a difference between the total revenue and total cost function. So, whenever there is a change in the total revenue and wherever there is a change in the total cost it generally affects the profit. So, profit function is basically the relationship between the profit, revenue and cost. Then we will discuss what are the general forms of function used in the economic analysis. So, one way we are clear that we use a functional form to understand the relationship between two types of variable and the variable typically in this case it both all the variables they are the economic variables. So, there are three type of function we used in analyzing the relationship between the variable. One is linear function, second one is the non-linear function and third is the polynomial function. Linear function is used when the relationship between the dependent and independent variable remain constant. Linear function is used where the relationship between the independent variable and dependent variable is not constant rather it changes with the changes with the change in the economic variable and polynomial function is getting used particularly those who have the various terms of measure for the same independent variable. So, we will check all this function in more details we are taking individually. So, in a linear function the relationship is linear the change in the dependent variable remain constant throughout for a one unit change in the independent variable irrespective of the level of the dependent variable. Whatever the change in the independent variable the change in the dependent variable is remain constant in case of a linear function. Suppose we are taking a demand function which says that q x is equal to 20 minus 2 p x. What does it signify? Each 1 rupees change in price the demand for commodity changes by 2 units because if you look at look at the second term of this functional form there is minus 2 p x. So, 1 rupees change in the price the demand for the commodity changes by 2 units. When you represent graphically the linear demand function is always a straight line because the change in the dependent variable remain constant, constant out for 1 unit change in the independent variable. So, this is just a hypothetical way to understand the linear demand where in the vertical axis we are taking price and horizontal axis we are taking quantity. So, if you look at when the price is changing the quantity demanded is also changing. So, initially when the price is 2 dollar the quantity demanded is 100 units. When the price increase from 2 unit 2 dollar to 3 dollar the quantity decreases by 100 units to 50 unit. So, if you look at the demand curve at each point it gives a price and quantity combination. Here quantity demanded is the dependent variable whenever there is a change in the price that leads to change in the quantity demanded also and if you look at in the percentage wise also when the price changes from 2 dollar to 3 dollar it is 50 percent change in the price and when the quantity demanded decreases it is decreases from 100 to 50 again this is a 50 percent decrease in the quantity demanded. So, 50 percent increase in the price is leading to 50 percent decrease in the quantity demanded and at this point the relationship between these two variables linear as it is constant whenever the price point change from one to another. Then we will discuss the non-linear demand function where the relationship between the dependent and independent variable is not constant it changes with the change in the level of independent variable. So, in the previous case the way we are discussing that 50 percent change in the price will bring 50 percent change in the quantity demanded. However, in case of non-linear demand function the unit of change it may not constant with each change in the price when the price change from 1 dollar to 2 dollar or 2 dollar to 3 dollar that may not necessarily the change the same kind of change in the level of the quantity demanded. So, if you are taking a non-linear demand function that is dx which is a function of price x. So, here x is the product px is the price of x and dx is the quantity demanded of x. Taking the functional form in a non-linear dx is dx is a px to the power minus b. Here a and b they are the constant minus b is the exponent of variable px constant a is the coefficient of variable px. If you simplify it further maybe you are taking a number term over here suppose dx is 32 px to the power minus 2 or maybe we can take just a reciprocal of this that 32 minus px square. So, in this case the demand function produce a non-linear or curvilinear demand function demand curve it means it is not a straight line the change in the independent variable is not constant throughout whenever there is a change in the price. So, this is an example of a non-linear demand schedule that how it changes when there is a change in the price. So, when price is 1 quantity demanded is 32, price is 2 quantity demanded is 8, price is 3 quantity demanded is 3.5 similarly for 4, 5 and 6 if you look at the trend the quantity demanded is going on decreasing when the price is increasing. But here the point is not to establish a negative relationship or inverse relationship between the price of x and dx the point what we are discussing here is that with each change in the price point the quantity demanded is the change in the quantity demanded is not remain constant. The change in the quantity demanded with respect to each price point is become different and this is a typical picture of a non-linear demand curve and when you plot this in a graph we generally get a curvilinear relationship which is in the form of a graph which is in the form of a curve we do not get a line we do not get a straight line which is generally the representation of a linear demand curve. So, this is the graphical representation of a demand curve and if you look at the different point of the demand curve the change in the quantity demanded is not remain same. So, when the if you are taking here the p is the price which is represented in the vertical axis and q is the quantity which is represented in the horizontal axis. When the price is changing from 100 to 80 the quantity demanded is increasing and again when it is decreasing from 80 to 20 the quantity demanded is again increasing. But if you look at the change in the price point from 100 to 80 and the corresponding change in the quantity demanded from may be 10 to 12 that is not remain constant and with the next change in the price point from 80 to 20 the there is a significant amount of change in the quantity demanded that is from 12 we need to 50 unit. So, in case of a non-linear demand curve even if the demand changes along with the change in the price point, but there is always a difference in the amount of change at different kind of price point. The third kind of function generally used in the economic analysis is polynomial function and what is polynomial function? The function contains many terms of the same independent variable are called the polynomial function. So, we consider a short run production function here where output is a function of the labor and output is represented as q and labor is represented as l. So, putting it in a functional form q is the function of l over here. The polynomial function takes a takes different type of functional form sometimes it is take a quadratic function sometimes it takes a q b function sometimes it is take a power function. So, taking the example of the same short run production function where q is the output l is the labor and a b c d r constant associated with the different coefficients it takes a quadratic function it takes a form by cubic function or takes a function of the power function. So, when it is become a quadratic function q is equal to a plus b l minus c l square your a b c r constant. When you take a cubic function then it is a plus b l plus c l square minus d l q where again a b c are the constant associated with the coefficient. When it takes a power function here it is a l to the power b here a and b is the constant and b is the coefficient associated with variable l. So, polynomial function may take a quadratic function polynomial function may take a cubic function and polynomial function also takes a power function. When you represent this polynomial function graphically with all this three type of function whether it is quadratic cubic and function. So, graphically if you look at a cubic function how this cubic function when the polynomial function takes a cubic function. Suppose we take l over here l is the labor q is the output. Now, the cubic function takes a this type of shape. Now, what is this curve? This curve is total product curve and total product is dependent on the output and the labor. So, if you are taking q over here and l over here cubic function take a form which is may be not a straight line not exactly a curve it follows a different kind of change at each change in the l. So, how this q and l they are related here l is the independent variable and q is the dependent variable. So, l is the whenever there is a change in the l that will bring in the bring change in the q. So, in this case this in a case of a cubic function l changes along with this q changes, but the change in the q is not constant with each change in the labor. Now, if you now if you take a case of a quadratic. So, with the same short run production function we take l over here in the x axis and q over here in the y axis. Now, it is a quadratic. So, just follow a there is no cyclical function over here or there is no much fluctuation here and this total product curve is this and this is a typical example of a quadratic function. Now, how graphically when you represent the power function of the power function of the polynomial function. So, the power can take any value the coefficient associated with b or the coefficient b associated with l it can take any form. So, if you remember the power function is q is equal to this is a to the l to the power b. So, b can take a value which is equal to 1 b can take a value which is less than 1 or b can take a value which is greater than 1. So, in this case if you represent graphically again taking the same formulation that here it is labor here it is output when we get the value of b equal to 1 it is a straight line the total product curve is a straight line when b is less than 1 we get a this kind of shape and when b is greater than 1 we get a this type of shape. So, if you take a power function in case of a polynomial function the power associated with the coefficient b can take any value may be it is sometime it is 1 sometime it is less than 1 and sometime it is greater than 1. So, it is whether it is 1 whether it is less than 1 and greater than 1 when you represent that graphically or when you represent that geometrically this is the shape what we get for the different kind of function. So, polynomial function takes the quadratic function polynomial function also can be represented through cubic function and polynomial function also represent through a power function and each time the value of b changes in time each time the graphical represent the changes depend on the value of the coefficient. Now, how do you find what is the degree of a polynomial function? So, degree of a polynomial function if you taking a functional form which is q which is equal to a plus b l minus c l square the highest power is 2. So, this is a polynomial function of degree 2 and a polynomial function of power 2 is also called a quadratic function. So, in order to identify what is the polynomial function it is always the highest power associated in this functional form. So, in this case the highest power is 2. So, the polynomial function of degree 2 we can say is having this functional form. So, polynomial function of power 2 is also this is also called a quadratic function. Let us take one more example in term of a cubic function. So, here what is the functional form? The functional form which q is equal to a plus b l plus c l square minus d l cube. Here the highest power associated with the coefficient is 3 and this is a polynomial function of degree 3. A function of power 3 is also called cubic function. So, in the previous example in the functional form the highest power is 2. So, that is why it was a quadratic function of degree 2 and in this typical functional form the highest power is 3 and that is the reason it is called as a cubic function because the function of power is 3. Then we will say we will see what is the degree of polynomial function when there is the when the polynomial function is in term of a power function. So, here in this case the functional form q is equal to a l to the power b. The range of power is between b greater than 1, b is equal to 1 and b less than 1. So, in this case except 0 it can take any power. So, it may be less than 1 in the negative form equal to 1 or it may be greater than 1. So, in this case b taking value of 0 is not possible, it takes any other value and this is the example of a power function under the polynomial functions. So, there are three type of function. One is linear, second one is non-linear and third one is polynomial. In case of polynomial function again we represent in term of a quadratic function in term of a cubic function or in term of a power function and every time the degree of it changes on the basis of the power associated with the functional form. The highest degree in case of a quadratic function is 2, the highest degree in term of a cubic function is 3 and the highest degree in term of a power function is greater than 0 it may take a negative value or it may take a positive value. Now, how to solve a polynomial function either through the factoring methods or through the quadratic formula and what is the property of a quadratic or a cubic equation when there is more than one solution. So, polynomial function can be solved by factoring methods and quadratic formula and by both these methods it can be solved and property of quadratic and cubic equation that it has more than one solution. Now, solving this quadratic and cubic equation we have two methods, one is factoring method and second one is the quadratic formula. Now, we will check one by one what is factoring method and what is cubic equation or what is the quadratic formula to solve this cubic equation. So, we will take a function that is y is equal to x square plus x minus 12. So, in the factoring method what is the first step? The first step is we have to set y is equal to 0. So, taking this x square plus x minus 12 is equal to 0. Now, what is the second step? We have to factor the equation. So, this x square plus x minus 12 can be also represented with x square plus 4 x minus 3 x minus 12 which is equal to 0. Now, simplifying again this we have this takes x plus 4 and x minus 3 which is equal to 0. We get two value of x over here if you simplify one is x is equal to 4 the second is x is equal to sorry minus 4 and x equal to 3. If you look at minus 4 has no meaning in economic analysis. So, basically you will go with the value of positive value that is x is equal to 3 and we solve this functional form with the value of x which is equal to 3. So, even if we are getting two value one is minus and second one is plus and typically since we are applying this in a economic analysis there is no maybe significance when we get a negative value of any variable. So, that is the reason we are ignoring the first value of x which is minus 4 and we are going with the second value of x which is equal to 3. So, this is so if you look at this solution is through factoring method. So, this is the solution with the solution of a polynomial function by using the factoring method. Now, we will check the second one through the quadratic formula. Now, what happens in case of a quadratic formula? The quadratic equation is set equal to 0 that is the first step and the equation is again the equation is factor for obtaining the two value of the variables x and y following the formula that is minus b plus minus b square minus 4 a 4 a c divided by 2 a. So, let us see how we can solve a polynomial function through the quadratic formula. Now, what is the functional form over here? The functional form is y is equal to x square plus x minus 2 l which is equal to 0 because what is your first step? First step is to set the quadratic equation or whatever the functional form we have to set that equal to 0. Now, what is the implication of this equation? Now, x is equal to how to factor it again to get the value of x because the first step is always to set it is equal to 0 and second we factor out this equation in order to get the two values of the variable x and y or if there is only one value the value of the x. So, if you are following then this is this minus b plus minus b square minus 4 a c by 2 a. Now, what is the implication of this equation? a is equal to 1 this formula taking this equation a is equal to 1 b is equal to 1 and c is equal to minus 12. So, this is the formula to factor out the equation this is the second step. The first step is to set this equal to 0 that is x square plus x minus 12 is equal to 0 and from this equation we get the value that is a is equal to 1 b is equal to 1 and c is equal to minus 1. Now, we will substitute this value of a b and c in case of in the quadratic formula. So, this is x is equal to minus 1 plus minus 1 square 4 a c and 2 a. So, this is 2 and 1. So, x is equal to minus 1 plus minus 49 root divided by 2 which you simplify again this minus 1 plus 7 by 2. So, in the previous case once we have identified the value of a is equal to 1 b is equal to 1 and c is equal to minus 12 we will put this value into the quadratic formula and we are getting this is the value of x. Now, this is 1 plus minus it means x is having 2 values here which satisfies the quadratic equation because this is minus was plus minus 7 plus minus 1. So, this is the value of x. Now, this is 1 plus minus it means x is having 2 values here which satisfies the quadratic equation because this is minus was plus minus 1 which is divided by 2. So, we will get 2 value of x and it satisfy the quadratic equation. Now, so if we take the first value that is x is equal to minus 1 plus 7 by 2. So, we get x is equal to 3 and if we take second value that is minus 1 minus 7 divided by 2 then we get x is equal to minus 4. So, we are getting 1 negative value we are getting 1 positive value even if this is negative still it is satisfying the quadratic equation. So, we have 2 value 1 is positive 1 is negative. So, basically we ignore the value which has some negative which is with negative sign and we always go for the positive sign value because it makes some sense in the economic analysis when we go for the positive value. So, if you remember the previous solution what we did through the factoring method also we got 2 value of x 1 is 3 and second one is minus 4. So, whether you solve the quadratic or cubic equation or whether you solve the polynomial function either by taking factoring method or by taking the quadratic formula you always get 2 values of x and on the basis of the value whatever we get for x on that basis we decide which value to take for the further analysis and which value to ignore over here. So, polynomial function can be solved by using by 2 methods one is the factoring methods and second one is the quadratic formula and we get same value for the x taking any specific formula either through the factoring method or through the quadratic formula. Then we will discuss what are the concept of the slope over here. Now, what is slope if you remember your marginal analysis what we discussed before may be few session or may be the last session the marginal change is always whatever the change in the dependent variable due to 1 unit change in the independent variable. So, slope is to measure the relationship between the marginal change in the 2 related variable and it can be also defined as the rate of change in the dependent variable as a result of change in the independent variable. So, through marginal analysis we know whether the dependent or the independent variable how it changes when there is a change in the 1 variable when there is a change in the independent variable how it leads to change in the dependent variable that we know through the marginal analysis. But slope through that we can measure what is the exact nature of the relationship whether they are positive related whether they are negatively related and we also we can also quantify the change that what is the percentage change or what is the amount of change that is taken place in the dependent variable due to change in the independent variable. So, geometrically if you look at what is a slope it representing the relationship between 2 variables in a line in case of a linear relationship and or a curve in case of a non-linear relationship. So, the slope of the line or the curve shows how strongly or weakly 2 variables are related. So, in order to find out the slope we represent graphically the relationship between 2 variables. So, if the 2 variables are linear variables linearly related we get a line, if the 2 variables they are non-linearly related we get a curve and slope generally says that how strongly or how weakly these 2 variables are related to each other. The steeper is the curve or the steeper is the line the weaker is the relationship implication for this is that they are not strongly related or they are not related if there is a steeper line or steeper curve it means there is no change in the no much change or no significant change in the dependent variable even if there is a change in the independent variable. However, if the curve or the line is more flatter or it is it is become more flat it is it signifies that there is a stronger relationship between these 2 variable or we can say if there is a small change in the independent variable it leads to a greater change in the dependent variable. So, when 2 variables they are represented through a line or a curve the slope measures the change between these 2 variable what is the amount of change what is the nature of change or in the other word we can say they can quantify whatever the relationship between these 2 variable. So, if it their steeper they are not much related if they are flatter then they are related to each other. Now, taking the typical example of a demand function over here or demand curve over here the slope is the ratio in the change in the dependent variable and change in the independent variable. So, if you look at it case of a linear demand curve we always get a straight line demand curve in case of a non-linear demand curve we get a curve. So, with respect to demand curve what is the slope? Slope is the ratio of change in the dependent variable D to the change in the independent variable. So, movement down the demand curve it gives the decrease in the price and if it is of the demand curve it is the increase in the demand and the ratio of the change in the price and the change in the demand gives the slope of the demand curve. So, if you know demand function is a function of price. So, price is independent over here and demand is dependent over here. So, how to measure the slope over here? The change in the demand due to change in the price that becomes the slope of the demand curve because this slope measures the change in the demand curve due to change in the price. So, in this specific case the slope is the change in the demand due to change in the price. So, we will see how the slope generally we get slope in case of a linear demand curve and increase of a non-linear demand curve. Suppose we take the example of a linear demand function that is d x is equal to 20 minus 2 p x this is a demand function. Now, when you find out the demand curve or how to find out the demand curve from this demand function. So, let us say this is 2, this is 4, this is 6, this is 8, this is 10, 12, 14, 16, 18 and 20. And similarly here we can say 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 10. So, when the price is 6 suppose we say the quantity demanded is 8, when the price is 5 the quantity demanded is 10, when the price is 3 the quantity demanded is 14. So, price is 6, quantity demanded is 8, we get 1 point of the demand curve, then price is 5, quantity demanded is 10, we get the second point of the demand curve, price is 3, quantity demanded is 14, we get the third point of the demand curve. If you join this 3 point we get the demand curve. So, maybe this is point j, this is point k and this. Now, what is the demand curve showing over here? If it is demand for x suppose what is the demand curve showing over here? This is the change in the price of x and the consequence change in the quantity demanded of x. Price we are considering here, quantity we are considering here. So, if you look at y axis is p and q is represented in x axis. So, this demand curve is essentially showing the relationship between the change in the quantity demanded due to change in the price. So, suppose the initial price as we mentioned initial price p x is equal to 6. So, the quantity demanded is 8 with respect to that. Now, suppose p x decreases from 6 to 5, quantity demanded increases from 8 to 10. So, this is what? This is the change in the p x and this is the increase in the d x. So, this is minus because there is a decrease in the price and this is positive or this is plus because there is a increase in the quantity demanded. So, when price of x decreases from 6 to 5, quantity demanded increases from 8 to 10. So, what is the value of del p x? Del p x is equal to minus 1 because it is changing from 6 to 5 or we can say may be it is from 5 to 6 then it become 1. Now, this del d x is from 10 to 8 because this is the change in the x. So, del d x is 2. So, p x is the independent variable and d x is the dependent variable. So, to change in the p x there is a change in the d x. So, given the value of del p x is equal to 1 and del d x is equal to 2. What is the slope of a straight line demand function? The slope of a straight line demand function is the ratio of del p x by del d x. So, this is the slope of the demand function. So, del p x and del d x this is become the ratio and through this ratio we can find out the slope of a straight line demand curve between the point j and k. So, if you see the previous graph this is the point j and k. So, through this ratio we can find out the slope between the 2 point j and k and this becomes 1 by 2 which is equal to 0.5. So, given the demand function d x is equal to 20 by 2 p x and p x is equal to 6 and d x is equal to 8 initially. There is a change in the p x from 6 to 5 and change in the quantity demanded from 8 to 10 that leads to the change in the price of x that is del p x and change in the quantity demand or d x. So, which is 1 by 2 and the slope is 0.5. This slope if you look at since this is a case of a linear demand curve the slope is constant throughout the demand curve. Now, suppose we consider that if price of x decreases from 5 to 3. This is again the change in the price of x. The cost of the demand quantity demanded changes from 10 to 14. So, this is 10, this is 14, this is the amount of change in the quantity demanded of x. So, if price decreases from 5 to 3 and quantity demanded increases from 10 to 14. We will find out what is del p x over here and what is del d x over here. So, del p x is 3 minus 5 that is minus 2 and del d x is 14 minus 10 that is 4. So, in this case when you identify what is the slope between this two points then this is again the ratio of del p x by del d x and which is again 2 by 4 and we are getting a value which is 0.5. So, in case of a linear demand curve you get a constant slope throughout all the point of the demand curve because the change in the dependent variable remain constant with respect to change in the independent variable. Next, we will see what is the how we measure the slope of the non-linear demand function. So, non-linear demand function let us take a functional form that is d x is equal to 32 p x minus 2 or we can say this is 32 by p x square. Now, in case of a non-linear demand curve the slope of the curve can be measured between two points between any two points and then we can compare what is the slope between these two points. What is the essential demand essential difference between a linear demand curve and a non-linear demand curve? In case of linear demand curve the change in the dependent variable remain constant throughout the entire demand analysis or entire analysis period. But in case of a curvilinear or in case of a non-linear demand curve the dependent variable changes in a cyclic manner or in a different proportion at each point of the demand curve. That is the reason here it is necessary to measure the slope between two different point and again compare whether the slope is remain same or slope is decreasing or slope is increasing or to identify what is the trend of the slope between different points of the demand curve. We take p x on the vertical axis and d x on the horizontal axis. So, here we get 1, 3 or may be we can 1, 2, 3, 4, 5, 6, 8, 10, 12 so on. And in case of p x we can say this is 1, 2, 3, 4, 5, 6 and 7. When price is 5 the quantity demanded is somewhere between 1 and 2. So, let us say this is 1.3. When price is 4 the quantity demanded is 2. And when price is 5 the quantity demanded is somewhere between 1 and 2. So, let us say this is 1.3. When price is 4 the quantity demanded is 2. And when price is 4 the quantity demanded is 3. Then price is 3 the quantity demanded is 3 and or may be say it is somehow 3. If it is 3 then this is 3.5. And when price is 2 then the quantity demanded is 8. Basically if you join this point suppose this is point A and this is point B and this is point C and this is point D. If you join all this 3 point all this 4 point rather we get a non-linear demand curve. Now, how we will identify there how we will measure the slope between 2 point. Now what is the slope between point A and point B? Now what is del p x over here? Del p x is the difference between 4 and 5 that is the change in the price that is 4 and 5. So, from point A to point B now what is the slope that is del p x by del dx. What is del p x difference between 4 and 5? So, that comes to minus 1 and what is the difference in the quantity demanded that is the difference between 2 and 1.3. So, that comes to 0.7. Now what is the slope over here? The slope over here is minus 1.43 that is between point A and point B the slope is 1.43. Now what is the slope between point C and point D? So, between what is the change in del p x? The change del p x is between price 2 and price 3. So, this is minus 1. What is the change in the demand? The change in the demand is between 8 and 3.5. So, that leads to 4.5. So, this comes to 0.23. So, if you look at in a non-linear demand curve the value of the slope changes or the value of slope is not constant in all point of the demand curve. So, when we measure or when we calculated the slope between point A and point B we got a figure which is 1.43 and when we calculated the slope between point C and point D the value of the slope is 0.3. So, we can say that the slope of the non-linear demand curve is different between the different points. Now when you measure the slope at a point of the curve what may be the limitation or what may be the constant over here? In case of a non-linear demand curve what we do? We calculate the slope at two different point taking the change in the price, taking the change in the quantity demanded and we then measure the value of the slope. So, what are the limitation over here? When you measure the slope at a point on a curve this method may not be reliable because particularly in this case when the change in the independent variable is large because slope is different from any set of two points within the chosen two points of the curve and the method is not much of help in case of a optimum solution to the business problem that has to be found because of an optimization problem may involve a polynomial function. So, measuring slope particularly for a linear and non-linear it is possible when it comes to polynomial it is basically difficult to use the same method to measure the slope and that is the reason the difference between two variables also sometimes so large that it is difficult to do analysis by measuring the slope in this way. That is the reason there is a technique of the differential calculation has come into existence in order to understand the margin in order to measure the marginal change in the dependent variable due to change in the independent variable particularly when the change of change in the independent variable approaches 0 and the measure of such marginal change in generally known as the derivative. The derivative of a dependent variable y is the limit of change and y when the change in the independent variable x approaches 0. So, because of the limitation to measure to measure slope at a point in a curve the technique of differential calculus generally comes into picture. So, differential calculus is generally used to find an optimum solution to the problem and this is used in the derivative of a constant function, derivative of a power function, derivative of a function of the sum and difference of the function. Function is a product of two function derivative of a quotient derivative of a function of function. So, we will check individually each function and how we use differential calculus over there, but before that we will see that how we can find out the differential calculus or how we can represent the differential calculus graphically. So, we are considering y is the dependent over here and x is the independent over here. So, when we represent this in a graph we take a function that is y is equal to function of x, x 1, x 2, this is y 1, this is y 2. Now, what is the change in the x that is change in the x from x 1 to x 2, this is the change in the y from y 1 to y 2. So, this is point a and this is point b. Now, when x increases from x 1 to x 2, y increases from y 1 to x y 2. So, point a shift from, so demand function shifts from point a to point b. So, in here what is del x now, del x is x 1, x 2 and del y is y 1, y 2. How we will identify what is the slope of this function? So, slope of this function is del y by del x which is y 1, y 2 by x 1, x 2. So, when the change in the dependent and independent variable is very small, the slope can be calculated from the method of the differentiation or the method of the differential calculus. Now, we will take the same example here, we have just taken a general function. Now, we will take a function specifically to the demand function to understand this differential, how differential calculus is being used in order to calculate the marginal change or in order to measure the changes between the two variables that is dependent variable and the independent variable. So, let us take a demand function that is dx is equal to 32 px to the power minus 2. So, this is a demand function, we will use the differential calculus to find out the slope. And when this differential calculus is required or when this differential calculus is helpful, when the change in the independent and dependent variable is very small, it is difficult to find out the value of slope by the formula what we discussed earlier. So, if you are taking the first order derivative equal to 0, then it will become del dx with respect to del px and so this comes to minus 2, 32 px minus 3. So, this comes to minus 64 by px cube. So, if we will take the reciprocal of the above equation, then this is del px and del dx and this is minus px cube by 64. So, if you take this point in the point b considering this function in point b, what we did the graphical representation earlier and substituting price is equal to 4. So, taking this and substituting price is equal to 4. Now, what will be the value of this equation that is minus 4 cube by 64. So, this is minus 64 by 64 which is cube equal to the minus 1. So, the slope of the demand curve using the differential calculus both by the tangent method and by the differential calculation this is 1. So, in this case what we did, we took the demand function, the same demand function what we did earlier by taking the general tangent method to understand the slope and we found the value of the slope is equal to minus 1. Now, we have taken a different formula or may be different methods that is the rule of differentiation or the difference calculation or derivative to understand or find out the slope and following the differential calculus taking the first order derivative and putting the value of p, we got the value of slope which is equal to minus 1. So, if you remember in case of a tangent method also the value of slope we got as the minus 1. So, whether we follow the tangent method or whether we follow a differential calculus method taking the same demand equation the slope becomes equal. The only difference here is that we cannot use the tangent method with all changes in the dependent or independent variable. When the dependent and independent variable the change related to them is small in that case only the differentiation or the differential calculation method can be useful. So, in the next session we will look at what are the rules of differentiation and as we discussed that how we deal the derivative of a constant function power function function of sum and differences of equation, quotient then power function everything we will discuss in the next class related to the rule of derivatives or the rule of differentiation.