 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 one unit change in the independent variable. So, slope is to measure the relationship between the marginal change in the two 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 one 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 two 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 two variables are related. So, in order to find out the slope, we represent graphically the relationship between two variables. So, if the two variables are linear variables linearly related we get a line, if the two variables they are non-linearly related we get a curve and slope generally says that how strongly or how weakly these two 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 become more flat, it signifies that there is a stronger relationship between these two variables 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 two variables they are represented through a line or a curve, the slope measures the change between these two variables, 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 two variables. So, if it they are 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 in 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 up 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, 13, 14, 15, 15, and 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 consequent 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 maybe it is from 5 to 6 then it become 1. Then 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 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 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 demanded 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 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 2 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 2 points between any 2 points and then we can compare what is the slope between these 2 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 2 different points 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 maybe we can 1, 2, 3, 4, 5, 6, 7, 8, 7, 4, 3, 4, 5, 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 3, the quantity demanded is 3 and or maybe 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, 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 d x. What is del p x different 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 in 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 reach 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 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 equation 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? So, 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 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 d x is equal to 32 p x 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 d x with respect to del p x. So, this comes to minus 2, 32 p x minus 3. So, this comes to minus 64 by p x cube. So, we will take the reciprocal of the above equation, then this is del p x del d x and this is minus p x cube by 64. So, if you take this point in the point b, considering this function in 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 q by 64. So, this is minus 64 by 64 which is keep 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 after 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 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 question then power function everything will discuss in the next class related to the rule of derivatives or the rule of differentiation.