 Welcome to the fifth session of managerial economics. We are discussing the first module of managerial economics that is introduction to managerial economics and in this topic if you remember in the last session, we discussed about the relationship between economic variables. We termed them in term of linear, non-linear. Then we discussed about that how these variables are related and what is the method to capture the relationship. So one was what we discussed in the last class is slope. Slope is basically it captures the change in between the dependent variable and the independent variable. But when the change in the independent variable is very small, in that case the general method to calculate slope or just finding the relationship through the slope sometimes does not serve the purpose. And that is the reason we introduce the concept of differentiation and differentiation is a method, differentiation is an approach through which we can calculate the or capture the change in the dependent variable due to change in the independent variable. So in the last class we have taken just a general functional form and identified the differential calculus method. In this class what we will do? We will take some different kind of function like constant function, power function, derivative of function of sums and differences of function. Derivative of function is a product of two function, derivative of a quotient and derivative of a function of a function. And also we will take a function where there are multi variables are there. So we will just check one by one how the differentiation is generally used or how through differentiation how we calculate the relationship between the two different variable independent and dependent variable in the different kind of functional form. So we will start with the constant function and where if you look at the functional form is y is equal to it is a function of x. So y is a function of x and which is equal to a. Now what is a over here? a is the constant. Now if we will take a first order derivative of this functional form. In this case what is the first order derivative? We have to take the derivative with respect to x and which will come as 0. Why it will come as 0? Because it is a function of x and that is in the form of a constant. And when you are taking the first order derivative with respect to a constant the derivative the value of the derivative will be equal to 0. So whatever the value of x, so whatever the value of x y remain constant here. There is no change in the y because y is represented in term of a function and function is a constant over here. So maybe if you modify this bit and if you put a value suppose you take a functional form y is a function of x and which is equal to 500 this is the value of the intercept this is not the value of the slope here. So in this case whatever may be the value of x there is no change in the y, y remain constant because this is a derivative of a constant function. Now we will discuss the second kind the derivative of a power function. So here the functional function form has a power on it. Suppose here again y is the dependent variable x is the independent variable it is a y is a function of x and which may be a x to the power b. Now here there are two constant one is a another is b. So y is a function of x which is a x to the power b and in this functional form we are getting two points one is a and second one is b. Now if you take a derivative of this how this will become what is the outcome over here del y with respect to del x that will be b a x b minus 1. Now let us give a number to this functional form. Suppose take a functional form where y is equal to 5x q. So x to the power q. So in this case y is a function of x and the x has also a power on it. So now what is the derivative or how we can check the relationship del y to the power del y with respect to del x. So that will come as the 3 5x 3 minus 1. So this is 15x square. Similarly you take one more functional form where y is equal to 4x square. Now how we will find the differentiation over here. So del y by del x is equal to 4 then 2x 2 minus 1. So del y by del x is 1. So this is 8x. Similarly suppose we take one more functional form which has also a power on it. So if you look at suppose we take y is equal to 2x x to the power 1 over here. Now if you take del y by del x then this is this comes to 1 multiplied by 2 x to the power x x then 1 minus 1. So 2x to the power 0 which is equal to 2. Now suppose take one more functional form in this category of derivatives like this is the derivative of a power function. So let us take y is equal to x. Now what is del y by del x del y by del x is equal to x 1 minus 1 which is equal to 1. Similarly if you take one more functional form which has a negative power y is equal to 5x minus 2. So this is del y by del x minus 2 5x minus 2 minus 1. And this comes to minus 10 x minus 3. So this is how we solve when we get a power function and what is the significance of this power function? The functional form has a power and it can take any value that is from less than 0 or that may be the greater than 0. Now we will discuss about the third category of the derivative that is derivative of a function of some differences of functions. If you will find it is not only a single variable there is also a summation added to it. So if you take an example where y is a function of x and also it is a g is a function of x. So y is dependent on this x function of x and also function of this x. So in this case how we take the derivative? In this case the derivative is del y by del x is equal to del fx with respect to del x plus del gx with respect to del x. Similarly now if you give a numerical term to this, suppose we take a functional form y is equal to 5x plus 2x cube. So now taking the first order derivative del y by del x is equal to 5x 1 minus 1 plus 2 multiplied by 3x 3 minus 1. So this comes to 5 plus 6x square. Now what happens if it is not a case of addition, if it is there is a differences it is not a sum rather it is a differences of function. Now we will take another functional form in order to show that when there is a subtraction or when the y is dependent on x and the functional form has a subtraction of the functional form has a differences between two variables how to get the derivatives of this. So let us take here that y is a function of fx minus gx. Let us keep a numerical value to this. So y is equal to 5x square minus 2x 4. Taking the first order derivative del y by del x is equal to 2 5x 2 minus 1 minus 4 2x 4 minus 1. So this comes to 10x minus 8x cube. So it is like if the y is dependent on a function which has the addition or which has the subtraction in that case we have to take the partial derivative with respect to both the variable. So in this case like it is fx and in the gx. Now suppose apart from these two variables let us add a constant also in the functional form. Let us take a functional form where y is equal to 4x cube minus 3x square minus 3x square plus 3. So now how we will get the derivative over here. So del y by del x is equal to 3 4x 3 minus 1 minus 2 3x 2 minus 1 plus 0 because the first order derivative of a constant would be always equal to 0. So this comes to 12x square minus 6x. So in case of fx derivative where the function is of sums or the differences we get the functional form in term of the negative value like there is a differences between two variable or there is a positive value there is a there is a summation between these two variables. Now let us check what is the next kind of a function. So this is a derivative of a function as a product of two function. So in the last case we took care of summation we took care of the subtraction. Now we will say that when the derivative of a function where the function is as a product of two function. So in this case how the functional form will be the functional form will be y is function of x and gx. So the functional form is product of two function that is fx and gx. Now how to take the derivative over here? Here we take the derivative keeping others as constant and in the second part we take the derivative of the second part keeping the first part as the constant. So in this case this is fx dgx dx plus gx dfx dx plus dx dx plus dx dx plus dx dx dx plus dx dx dx plus dx dx. So now let us take a numerical function to get more clarity. Suppose we say that y is equal to 5x square 4x plus 3. So here y is dependent on two function one is 5x square second one is 4x plus 3. So this is a case where the function is a function of two other function. So in this case how to solve or how to find out the derivative? So in this case 5x square plus multiplied by 4 that is the derivative of del 4x plus 3 plus 4x plus 3 as constant and taking the derivative of 5x square. So if you are taking the derivative of 5x square how much it will come? It will come as 10x. In the first case we have to keep 5x square as the constant and we have to take the derivative of 4x plus 3. So which will come as 4. So the first case fx is constant the derivative is with respect to gx. Second the gx is constant the derivative is with respect to fx. So in this case if you again simplify this then this comes to 20x square plus 40x 40x square plus 30x. So which come to 60x square plus 30x. Now let us take another functional form and where the function is again as a function of two other function. So it is a product of two other functions which serve as a functional form for the independent variable where the value is defined by the independent variable. Now suppose take a numerical example where y is a function of x square plus 2x square sorry x cube plus 2x square plus 3. So this is one function and the other function is 2x square plus 5. So there are two functions over here. So taking the first derivative del y by del x is equal to so in this case the first one will be constant now x cube plus 2x square plus 3 multiplied by the derivative of the second function. So the derivative of the second function is 4x. Similarly now the second term 2x square plus 5 will be constant and we have to take the derivative of the first function that is 3x square plus 4x. So simplifying this we get 4x4 plus 8x cube plus 12x. For the first one for second one it is 6x4 plus 8x cube plus 5x square plus 20x square plus 20x square that comes to 10x4 plus 16x cube plus 5x square plus 32x. So when the functional form is a product of two function in that case basically we take the derivatives by keeping the first factor first function is constant and taking the derivative of the second function and in the second case we keep the second function as the constant and we take the derivative of the first function when the functional form is product of two functions. Next we will check how we generally solve or how generally find the derivative of a quotient. So here the functional form is y j function of x. So let us write a functional form this is a general functional form but when you take that this is derivative of a quotient here the functional form involve a quotient of two function that is fx and gx. In this case how to get the derivative what is the formula to calculate the derivative. So del y by del x is equal to gx multiplied by derivative of fx with respect to del x in the first case gx is constant the derivative of the fx and second case fx is constant derivative of gx divided by gx to the power whole square. So when the derivative involves a quotient or the function of a quotient two function of a quotient in this case the derivative is derivative generally we calculate derivative by following this formula that is gx multiplied by d derivative of fx with respect to x second term minus second term remain constant fx remain constant we take the derivative of gx with respect to dx as a whole this divided by gx to the power if it is the whole square of this. Now let us take a functional form to understand this in a numerical term. So y is a function of 5x plus 4 and 2x plus 3. So in this case how we will get the derivative del y by del x is equal to 2x plus 3 then we take the derivative of this that is 5x plus 4 that comes to 5 minus we have to keep fx as constant. So 5x plus 4 is the constant we have to take the derivative of 2x plus 3. So that comes to 2 as a whole 2x plus 3 whole square of this. So if we simplify this this comes to 10x 10x plus 15 minus 10x plus 8 divided by 2x plus 3 square. So this is equal to 7 by 2x plus 3 square. So in case of a functional form when there is a it is a quotient of 2 function in this case we generally follow a formula which keep one function constant by taking the other function derivative of the other function and finally it is divided by the functional form of the second function sorry the whole square of the second function is the divided by the whatever the derivative and the constant of the other two function. So now next we will see the last category that is when the function is a function of function in that case how the derivative how to find out the derivative. So we discuss about a power function, we discuss about a constant function, we discuss about a function which has some and differences differences, we discuss about a function which has the product and we discuss about a function which has in the quotient form. So now we will discuss of a functional form which is function of a function. So now takes a functional form y is a function of u and u is a function of x. So in this case how to find out the derivative del y by del x is equal to del y by del u and del u by del x. So here the derivative is also two term one is del y by del u and second one is del u by del x. So let us take again a numerical functional form to understand this more. So y is equal to u q plus 5 u and u is equal to 2 x square. So del y by del u is equal to 2 x square. So del y by del u is equal to 3 u 3 minus 1 plus 5 and del u by del x is equal to. So before finding this del u by del x let us simplify it more this del y by del u. So this comes to 3 u square plus 5. Now what is del y by del u? So u is equal to 2 x square as we know this is the functional form u is equal to 2 x square. So if you put this value over here so then 3 2 x square to the power again square plus 5. So which comes to 3 2 x square again 2 x square plus 5. Simplifying it further so 3 multiplied by 2 x square again multiplied by 2 x square. So if you simplify this again so this comes to 12 x 4 plus 5. So this is what this is our del y by del u. This is the first part of it. Now coming to the second part, second part is derivative of u with respect to x. Why the derivative is u? Derivative of u with respect to x because the u is in a functional form and u is here the dependent variable the value of the u depend on the value of the x. So taking the derivative of u with respect to x we get 2 2 x 2 minus 1 which is equal to 2 x equal to 4 x because u is equal to 2 x square. So del u by del x is 2 multiplied by 2 x 2 minus 1 which comes to 4 x. So now taking this del y by del x which is again a function of del y by del u and del u by del x. So that comes to 12 x to the power 4 multiplied by 5 multiplied by 4 x. So this comes to 48 x 5 plus 20 x. So what is happening over here? Here the functional form associated with the dependent variable has two function. It is a function of function. So the variable is directly not getting generated from the functional form rather because the functional form is again a function of the other functional form. So in that case generally we first find out the value of y which is with respect to u. Then we find out the value of u with respect to x and then finally find the derivative of y with respect to x which is a product of derivative of y with respect to u and derivative of u with respect to derivative of x. So taking the same example it comes to 12 x to the power 4 plus 5 multiplied by 4 x which come to 48 x to the power 5 and 20 x. So in this case we discuss about a constant function, we discuss about a power function, we discuss about a function with some and differences, we discuss about a function which has the product, we discuss about a function which has the quotient value, we discuss about a function where the function is again a function of some other variable. Now let us analyze one more thing where the functional form has several independent variable. So here we generally take a functional form where y is a function of x. Now we will introduce a case where the value of y is not only getting valued in the form of the value of x rather there are several other independent variable those who are deciding the value of y. We will take a value, we take a functional form with the multivariable and then we will see how to find out, how to use the differential calculus or how to find out a derivative, the first order derivative in order to understand the relationship between the independent variable and the dependent variable. So in this case how the functional form will look like. So let us take a case of a demand function because in case of demand function the primary variable it affects demand each price but if you look at there are several other variables those you affect the demand for the product. So we will take a case of a typical demand function where we assume that demand is not only influenced by the price, there are certain other variables also influence the demand and we will see how we find out the relationship by taking the derivative with respect to different variable. So here if you are taking let us say the demand function that is demand for x, it is dependent on price of x, dependent on price of the substitute good, dependent on the price of the complement goods, dependent on the income, dependent on the advertisement and dependent on the taste of the consumer. So here P x is the price of product, P s is the price of substitute good, P c is the price of complement good, y is the income, A is the advertising expenditure and T is the taste of the consumer. So, similarly there are some other function what gets used in economics typically, where the dependent variable is dependent variables are dependent on many independent variable not only one independent variable. So, this is one example is demand function. Similarly, there is also example of production function. So, if you take a production function it is always a function of capital and labour. So, at least typically in the long run the function is not only capital or the function is not only labour rather it is a function of both capital and labour. So, if in this case if you consider Q is the output which is dependent on the capital and the labour in this case the value of output dependent on two independent variable that is value of capital and value of labour. Similarly, we can take a cost function where it is a function of capital, where it is a function of the rent what we pay for the capital, it is a function of the whatever the price we pay to the labour and it is a function on the whatever the wages and the salary we are paying to the wages. So, in this case particularly when the functional form has many independent variable in this case generally we use the partial differentiation in order to understand the relationship between the two variable that is x and y. Now, let us take a functional form. So, here y is a function of x Q plus 4 x square plus 5 z square. So, we have two variables here those were influencing y, y is the dependent variable and x and z they are the independent variable. So, now how we will find the derivative or how we will establish the relationship between y and x and x and z. So, the first thing what we will do we will find out the derivative with respect to x keeping the j keeping the z constant and in the second case we will find out the derivative with respect to z keeping the x as the constant. We need to do a partial differentiation with each variable in order to understand their relationship with the dependent variable and we cannot take the derivative of both the variables simultaneously rather we will keep one as the constant and the derivative of the other as in order to understand the relationship. Now, taking the same functional form where y is a function of y is a function of x Q plus 4 x square plus 5 z square we will first find out del y with respect to del x. So, this will come as 3 x 3 minus 1 plus 4 z. Similarly, how we will find the second one that is del y and with respect to del z which is 4 x plus 2 5 z 2 minus 1. So, this is 4 x because z is constant. So, in this case we get 4 x plus 2 x 2 multiplied by 5 z that is 2 minus 1. So, that comes to 4 x plus 10 z. Similarly, if you take a functional form suppose y is equal to a x to the power b and z to the power c. So, in this case this is a case of your power function where the y is dependent on x and z. So, in this case how we find the derivative? Derivative of y with respect to x keeping the value of z constant that is b a x b minus 1 z c and second is del y and del z keeping the value of x remain constant. So, that comes to a x b c z c minus 1. So, when you have a functional form which has many variables and many variables particularly in the independent variable category and where the dependent variable is dependent on many independent variable. In that case in order to find out the derivatives first we need to keep one variable constant taking the derivative of y with respect to the other variable and second case we need to keep the other variable as constant and taking the derivative of the y with respect to the present variable. So, with this we almost completed that what kind of derivative or how the derivative is used in case of different functional form. Whether the functional form is the constant it has the power function or may be it is addition, subtraction, product quotient or a particular function of a function.