Lec 10: Rules of differentiation

Hello, and welcome to another lecture of the course Mathematics for Economics Part 1. So, the topic that we have been covering is called differentiation. It is quite a long topic; we have covered some parts of it, and the particular subtopic within this topic that we were talking about in the last lecture is called limits. So, let us go to that particular subtopic.

So, here is the definition of the limit. Suppose the function f(x) is defined for all values of x near a, but not necessarily at x = a, then f(x) is said to have a limit equal to A as x tends to a if f(x) tends to A as x tends to a. So, basically, what one is talking about is that take any function. Let me draw a particular diagram to motivate the issue. So, if you have a function something like this and you have a particular value of x which is a, then this function, the value of the function approaches this particular value A as x approaches the value a. And we have examined one particular example which is like this: suppose f(x) has this particular form √(h + 1) - 1 / h. And we have taken values of h which are close to 0. At h = 0, the value of this function becomes undefined—something divided by 0 cannot be defined—but as you can see, as we are taking h very close to 0, 0.01, 0.1, 0.2, etc., etc., the value of the function approaches 0.5. And from the left-hand side also, as this should be minus, as h approaches 0 from the negative side, you have -0.05, -0.2, -0.1, -0.1; the value of this function is approaching the value of 0.5. So, therefore, by this sort of heuristic demonstration, we can say that the limit as h goes to 0, the limit of this function is 0.5.

Here, for a general case, this limit of this function is A, and where we can say that the limit does not exist. Suppose you take a function like this; so here it is defined, but suppose if you take x > a, then this function is actually starting from here. Then we say this limit is not there; the limit does not exist of this function at x = a because from the left-hand side, there you can make the value of the function as close to A as possible as x becomes very close to a; the value of the function becomes as close to A as possible, but that is from the left-hand side, but from the right-hand side, this is not true because from the right-hand side, you cannot make the value of the function as close to A as possible because as you can see that there is a gap. So, we say that the limit does not exist at this particular value of x = A. But suppose you take a kinked sort of shape, then can we say the limit exists? Suppose the function is something like this. Actually, here you can say that the function has a limit as x tends to a, then the value of the function tends to A, so the limit of the function exists at x = a. So, if there is a kink like this, then there is no problem, but if you have a gap like this, then there is a problem of the existence of the limit.

Now, there are some rules of the limits, so that is what we want to start in this class. Suppose this is given to us: that is, lim_x→a f(x) = A and lim_x→a g(x) = B, then what about the sum? What about the sum of f(x) and g(x)? As it turns out, the limit of the sum is the sum of the limits. On the left-hand side, you have the limit of the sum, that is, f(x) + g(x), and on the right-hand side, you have the sum of the limits, which is A + B. So, this seems quite intuitive. Similarly, the second rule is: if you take the gap, that is, the subtraction of f(x) and g(x), if you subtract g(x) from f(x) and take the limit of that, then the result is the same as if you take the limit of f and then from that subtract the limit of g; in both cases, x is tending to a. Thirdly, if you multiply two functions, so if you have f(x) and g(x) and take the product of f(x) and g(x) and take the limit of that as x tends to a, then the result is the same as the limit of f(x) multiplied by the limit of g(x), so that will be A multiplied by B. Fourthly, if you take the ratio of two functions, then what about the limit of that ratio? And this is equal to the ratio of the limits because on the right-hand side you have A / B. What is A? A is lim_x→a f(x), and B is lim_x→a g(x). So, the limit of the ratio is the ratio of the limits, and then you have an interesting property of power. So, you have f(x)^p/q, and then you are trying to take the limit of that, and the result is the same as you take the limit of f(x), which is A, and raise the power to p/q. So, these are the five sort of simple rules of limits.

There are some specific functions that one can take. Suppose it is a constant function, so suppose f(x) is the same; it is constant; suppose it is c, and you want to find out what happens to the limit as x tends to a, and the limit is the same as the value of the function because the value of the function is not changing; in the limit also, it is the same as the value. What about a very simple function where f(x) = x? So, this is like the 45-degree line in the first quadrant if I take x only non-negative values, and this is the 45-degree line, then the limit of this function as x tends to a will be a. The reason being that at x = a, the value of the function is a, so as x becomes very close to a, the value of the function is very close to a.

Now, we come to differentiation; we have already talked about the differentiation, but while talking about the differentiation, the idea of the limit was there. So, that is the reason why we had made this short sort of discussion about limits. Now, we are back to the discussion on differentiation, and let us talk about some rules of differentiation. As we know, the differentiation of any function can be done, and this is called the derivative of this function f(x) at any value x. So, this is called the derivative: lim_h→0 [f(x + h) - f(x)] / h. If this limit exists, then the function is said to be differentiable, and we have seen that the limit may not exist in many cases, and if the limit does not exist, then the function is not differentiable. The process of finding the derivative is called differentiation. So, there are two terms that we have defined here. One is a function can be differentiable; it may not be differentiable at a particular point, and this process of finding the derivative is called differentiation.

Suppose y = f(x) is the function, then the derivative is denoted by these notations. It is sometimes written as dy/dx, and it is also written as y' or sometimes it is called y prime, and it is sometimes written as f'(x); it is also called f prime of x. Now, what is being said is that it can be thought of as another function; so this can be shown very simply. Suppose you have y = x², then by using this definition of derivative, what you can say is that dy/dx = 2x. Now, this can be seen as a different function altogether; let us call this as Z, Z = dy/dx. Now, one can see that Z itself is a function of x. So, that is the simple point that is being made: that if you take the derivative of a function, then the derivative itself is a function.

So, now we can talk about another very simple function. Suppose f(x) = A, which is a constant, then what about the derivative? The derivative is actually 0; f'(x) = 0. What is the proof of this? This can be verified in two ways. You can think about the slope of the tangent to any horizontal line. So, here f(x) is what? f(x) is A; this value is A. Now, as we know, the interpretation, the geometric interpretation of the derivative of a function is that it is the slope of the tangent to the graph of the function. Now, if you have a horizontal line, then the line itself is horizontal, so its slope is 0. So, therefore, the derivative is also going to be 0; so that is one way to think about it. Another way to understand why the derivative should be equal to 0 is to go by the definition: f'(x) = lim_h→0 [f(x + h) - f(x)] / h. So, it got shifted a bit to the left. So, we know that f'(x) = lim_h→0 [f(x + h) - f(x)] / h, then we know f(x) in this case is equal to A. So, f(x + h) - f(x) will be equal to A - A, which is equal to 0. So, the numerator is 0, so this is called, if you remember, the Newton ratio. The Newton ratio becomes 0; therefore, it does not matter what limit I take; the f'(x) will be equal to 0.

Now, another function let us take which is a constant multiplied by a function of x. So, g(x) = A * f(x); f(x) is a function of x, obviously, then what should be the formula for g(x), the derivative of g(x)? Here, the derivative of g(x), which you are calling as g'(x), g'(x) is equal to that same constant A multiplied by the derivative of f(x); the derivative of f(x) is f'(x). So, any function multiplied by a constant, if I take that to be a new function, then the derivative of this new function is the constant multiplied by the derivative of that function.

Now, another important rule which is called the power rule. So, suppose you have f(x) = xⁿ, then what about f'(x)? And this is given by n * x^{n - 1}, and what is the proof of this? We can verify this by again using the definition of the derivative, which is lim_h→0 [f(x + h) - f(x)] / h. Now, in this case, what is f(x + h)? It should be this because f(x) = xⁿ, so f(x + h) should be equal to (x + h)ⁿ. Now, this term can be decomposed as xⁿ + nx^{n - 1}h + n(n - 1)/2, etc., etc. But whatever these terms are, the power of h should be 2 multiplied by something, then h³ multiplied by something, plus etc., etc., and the last term will be hⁿ. Now, from this, I am going to deduct xⁿ; that will be the numerator f(x + h) - f(x). So, from here, I have to deduct f(x), which is xⁿ. So, this term becomes nx^{n - 1}h + h² multiplied by something + h³ multiplied by something + ..., and then I have to divide this by h. If I divide it by h, then I have x^{n - 1} * n; h will get canceled, and then I have h multiplied by something + h² multiplied by something, etc., etc. Now, the limit of this will be: the other terms will vanish; all these terms will go to 0. So, you are left with only n * x^{n - 1}, so that is the final result: that if you take the derivative of xⁿ, then what you get is this n * x^{n - 1}.

Now, there are certain rules of differentiation. So, first, we are talking about the summation. So, suppose you have two functions f and g, and we know what their derivatives are, f'(x) and g'(x), then if I take the summation of them and call the summation as F(x), then the derivative of F(x) is equal to the summation of the derivatives, that is, f'(x) + g'(x). So, basically, the derivative of the summation is equal to the summation of the derivatives. Similarly, if I take two functions and deduct one function from another, so you have f, one function, and g, another function; f(x) - g(x) = G(x), suppose, and like before, we know f'(x) and g'(x) are the derivatives of f(x) and g(x), then G'(x) = f'(x) - g'(x). So, both these results are intuitive, and how can we prove that? We can actually start from here; this is the Newton quotient, and we know if we take the limit of h goes to 0 of the limit of the Newton quotient, then we get the derivative. So, if I use this and then I use the fact that f(x + h) will be what? It will be f(x + h) + g(x + h) for the first result. And similarly, f(x) is f(x) + g(x). And then we substitute these things back here, and that will give us this result by simplifying the expression; similarly for the second result.

So, here is an example of economics. So, you have, suppose, what is called the profit function, which is usually denoted by π(Q); Q is the output level. So, the profit function is expressed as R(Q) - C(Q), where R is the revenue and C is the cost. So, R(Q) is the revenue function; C(Q) is the cost function. Once the cost is deduced from the revenue, then we get profit; very simple. Now, suppose we want to want to find out what is π'(Q)? Then we use this rule here, and if we apply this rule here, then π'(Q) = R'(Q) - C'(Q). So, how can we interpret this? The interpretation is that marginal profit is equal to marginal revenue—this should be revenue—marginal revenue minus marginal cost. So, R'(Q) is called marginal revenue, and C'(Q) is called marginal cost. Now, one might be interested to find at what output level the profit is getting maximized. The producers could be thought of as maximizing profits. So, what output level is the profit getting maximized? That is a practical question. So, at the output level where profit is getting maximized, one can show—we can shall see it later—that level of output can be found out by setting π'(Q) = 0. So, if π'(Q) = 0, then the profit is getting maximized at that particular output level. So, this is a necessary condition; so if this is not satisfied, then the profit is not getting maximized, as we shall see later on. So, if you apply this rule, that is, π'(Q) = 0, then from this above, we get R'(Q) - C'(Q) = 0, that means R'(Q) = C'(Q). So, this is a necessary condition of profit maximization. At the output level where marginal profit is 0, marginal revenue is equal to the marginal cost.

We come to another rule which is called the product rule of differentiation. So, here if you take the product of two functions f and g, and suppose the product is called F. So, F(x) = f(x) * g(x), then interestingly, unlike what our intuition would tell us, f'(x) is not equal to f'(x) * g'(x). It is actually equal to this. So, F'(x) = f'(x) * g(x) + f(x) * g'(x). So, if we use the Leibniz notation, then this is how it looks like: d/dx[f(x)g(x)] = d/dx[f(x)] * g(x) + f(x) * d/dx[g(x)]. This is called the product rule. So, here is an example of how we can use the product rule. Oil is extracted from a well, and suppose it is given by x(t); the rate of extraction per day in barrels, so x(t) amount of oil is being extracted, whereas the price of oil is given by p(t), which is the price per barrel in rupees. Both of this, that is, x and p, they change with time; that is why they are functions of t. Now, the revenue per day in rupees will be what? R(t) = x(t) * p(t). The amount of oil that is being extracted multiplied by the price of oil; you get the revenue per day. Now, we can now use the product rule. So, we are calling it as Ṙ. So, Ṙ = dR/dt. So, this is the notation that we generally use when time is involved. Here, t is time. So, applying the product rule, what we should get is Ṙ(t) = ẋ(t) * p(t) + x(t) * ṗ(t). So, what is the interpretation of this? On the left-hand side, you have Ṙ(t), which is the change in revenue from oil extraction per day. As time changes, how much the revenue is changing. So, it comes from two reasons. First, if more oil is extracted per day, that is, x is changing, ẋ. Second, if the price of oil per barrel rises. So, if the price itself is rising, even if the production is remaining constant, revenue will rise. So, therefore, you have this ṗ term, and from this, we can actually divide both sides by R(t), which is the revenue, and we get this expression: Ṙ/R = (ẋp + xṗ) / R, and that simplifies into ẋ/x + ṗ/p. So, Ṙ/R = ẋ/x + ṗ/p. And in words, what does it mean? It means the proportional change, proportional rate of growth of revenue is equal to the sum of the proportional rate of growth of quantity and price. Remember, the derivative of R with respect to time, if I divide that by R itself, that is the expression for proportional growth of the revenue. So, therefore, the proportional growth of revenue is equal to the sum of the proportional rate of growth of quantity and price.

The product rule can be generalized to any number of factors on the right-hand side of f(x). What is meant by this is that suppose f(x) is, instead of just f and g, I take this as a(x) * b(x) * c(x). Suppose only three functions are involved, so you do not have two functions on the right-hand side, but three functions of x, then what happens to the product rule? Here, the product rule will be the following: you take a—this is the first term—so you are taking one derivative at a time. Out of the three functions, you are taking the derivative of one function at a time and keeping the others as functions themselves; they are not differentiated. So, you are going to get three terms on the right-hand side. The proportional growth rate of a variable would be likewise be the summation of proportional growth rates of individual factors. So, if you have something like this: F(t) = a(t) * b(t) * c(t), and suppose you want to find out this, the proportional rate of growth of F, then what is being said is this will be equal to this. So, it is basically the summation of three terms; each of these terms denotes the proportional rate of growth of each of these three functions on the right-hand side.

Here is another example where we can use these rules of the product rule. Suppose D(P) which is the demand for a good whose price is given by P. So, D is demand; P is price; D is a function of price. The revenue earned by the producer will be R(P), so R(P) is how much you are selling multiplied by the price of the good; that is revenue; how much are you selling? That is given by the demand because people will be demanding, and that is why you will be selling. So, R(P) = D(P) * P. Now, we can use the product rule here. So, R'(P) will be given by this; so what is the thing that is going on in the background? I have jumped one step. This is basically D(P) multiplied by the derivative of P with respect to P. The derivative of P with respect to P is 1, so nothing is coming here, plus P multiplied by the derivative of the demand function with respect to price, which is D'(P), P * D'(P), or we can take D(P) common on the right-hand side. And what we should get is 1 + P * D'(P) / D(P). Now, this term is an interesting term; let me write it separately; it is P * D'(P) / D(P). It has special significance; it measures the proportional change in demand with respect to the proportional change in the price. It is called the price elasticity of demand. So, what is this term, P * D'(P)? D'(P) is the change in the quantity demanded with respect to the instantaneous change in the price, and the whole thing is divided by the quantity demanded, which is D(P). So, this is called the price elasticity of demand. Now, there is something called the law of demand, by dint of which the D'(P) is negative, which means if price rises, the demand actually falls, so D'(P) is negative. So, the above expression is generally negative. The above relation establishes that if revenue rises or falls with respect to change in the price, it crucially depends on the values of price elasticity of demand. So, if I take the absolute value of this, it is negative, but if I take the absolute value, and if it is greater than 1, then this whole expression will be negative. So, that will mean that if price rises, then the revenue will fall, or if price falls, then the revenue will rise. So, whether the revenue rises or falls with respect to price, it crucially depends on this expression, the price elasticity of demand.

Now, we come to another rule, which is the derivative of a quotient. So, if f and g are differentiable at x, and suppose g(x) ≠ 0, then F, which is equal to f / g, it is differentiable at x. Now, if it is differentiable, then what is the derivative? So, that is given in the next line, so we have F(x) = f(x) / g(x), and F'(x) is given by this expression: [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]². In words, the derivative of a quotient is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator. This difference is divided by the square of the denominator; that is the explanation in terms of words. So, we take one example to actually grasp what is going on here; we take an example where f(x) = (3x + 7) / (x - 5). We have to find two things: f'(x) and f...

A dash of 2. So, if I can find out f dash of x, then that is a function of x. We have seen that the derivative itself is a function of x. So, to find f dash of 2, I just have to replace x by 2. So, the first task will be to find f dash of x. So, we use the quotient rule; we get this. So, in the denominator, you have the square of the denominator of the original function, which is (x - 5) whole square, and in the numerator you first have the denominator (x - 5) multiplied by the derivative of the numerator, which is 3, plus the numerator, which is 3x + 7, multiplied by the derivative of the denominator, which is equal to 1. And then we simplify this; it becomes 3x - 15 + 3x + 7, and that turns out to be 6x - 8, and obviously this is going to be divided by (x - 5) whole square. So, we have found out what is f dash of x, but there is another part, which is we have to find out what is f dash of 2. That is easily found out by replacing x by 2; so if we do that, then on the numerator you have 12 - 8, which is 4. And divided by (2 - 5) whole square, which is 9. So, therefore, it becomes 4 divided by 9.

Again, I talk about another example from economics. Now we know there is this thing called the cost function, C(x); x is the output, C is the cost. The average cost is given by C(x) divided by x. Basically, we are dividing the cost function by the quantity produced; that gives us the cost per unit of output, and that is called the average cost. Now, one might be interested to find out the derivative of the average cost, and here the independent variable is the output level. So, we have to take the derivative of the average cost with respect to the output level. Now, we use the quotient rule. So, this is on the left-hand side; you have the d/dx of C(x) divided by x, and if I use the quotient rule, I get this expression: x multiplied by C prime of x minus C(x) divided by x square. Now, let us simplify it further. So, if I simplify it a little bit, it becomes 1 divided by x, whole thing multiplied by C prime of x minus C(x) divided by x. In other words, the average cost rises if the marginal cost is greater than the average cost. Why am I saying this? Because this first expression, if you remember, this is called the marginal cost. We have talked about this before, and this itself is average cost. So, d/dx of the average cost is equal to 1 divided by x multiplied by marginal cost minus average cost. So, if it is the case that marginal cost is greater than the average cost, then the right-hand side is positive. And if the right-hand side is positive, then the left-hand side is positive, which means that as the output level rises, average cost is rising. Similarly, if marginal cost is equal to the average cost, then on the right-hand side you have 0, and if it is 0, then the d/dx of average cost is equal to 0; that means the average cost does not change; it remains the same. And finally, if the marginal cost is less than the average cost, then from the right-hand side you are getting a negative expression; marginal cost is less than the average cost; in that case, the left-hand side is also negative; that means the average cost is declining as the output level is rising. We will see that these things are quite important if one talks about what is called the theory of the firm.

Now we can also talk about proportional change. Here, suppose you have a function F(t), which is equal to m(t) divided by n(t). So, both m and n are functions of t. We use the quotient rule here, and we will get this expression: that d/dt of F(t) is equal to m dash of t divided by n(t) minus m(t) divided by n(t) multiplied by n dash of t divided by n(t). I have jumped some steps here, but this is what you are going to get by using the quotient rule. Now, from the above expression, then I divide both sides by F(t), and I will get this expression because F(t) is equal to m(t) divided by n(t), and then I multiply this term with the terms in the third brackets, and I will get this expression, and which means that F dash of t divided by F(t) is equal to m dash of t divided by m(t) minus n dash of t divided by n(t). Or, in other words, since F dash of t is equal to F dot t, so F dot t divided by F(t) is equal to m dot t divided by m(t) minus n dot t divided by n(t). So, if I express this in words: if a variable is the quotient of two variables, then the proportional rate of change of the variable is the proportional rate of change of the numerator minus the proportional rate of change of the denominator. This is the proportional rate of change of the numerator, and this is the proportional rate of change of the denominator. You take the difference of these two; you get the proportional rate of change of the ratio. This relation is again quite important in economic analysis.

So, for example, there is something called the real wage rate, so small w(t). It is the ratio of the money wage rate and the price level. So, suppose capital W is the money wage rate and capital P is the price level, then the real wage rate is defined by this: so small w(t) is equal to capital W(t) divided by P(t). So, the total amount of money people are getting as wages divided by the price level; that is called the real wage rate. Now, we can use the rule here that we have just got. Then we should get the following expression. In words, what we are getting is that the proportional rate of change of the real wage rate is equal to the proportional rate of change of the money wage rate minus the proportional rate of change of the price level, and this is not very difficult to understand if you think about it intuitively; the real wage rate of the worker, suppose, what does it mean? It means the purchasing power. We are dividing the amount of money that they are getting by the price level. Now, that real wage rate can rise because of two reasons. One is that the money that they are getting is rising, or it is possible that the price is declining. So, in both cases, their real wage rate will rise. So, that is why you have the proportional rate of change of the real wage rate is equal to the proportional rate of change of the money wage rate minus the proportional rate of change of the price level. So, it also says that suppose the actual money that the people get is constant; that is not changing, but the prices are rising; in that case, on the right-hand side, this will be equal to 0, where this is positive. So, the entire term becomes negative, which means that in that case, the real wage rate of the workers will be declining, and this is quite intuitive.

We shall start with this topic in the next lecture: partial differentiation. So, we have today covered the topic of differentiation proper; we have not talked about partial differentiation, and we have talked about different rules of differentiation; the summation of two functions; if we take the derivative of that, then what happens. If we take the product of two functions and take the derivative; if we take the quotient of two functions and take the derivative, then what are the expressions that one is likely to get. We would also talk about power rules; we talked about the derivative of a constant term, etcetera, etcetera, and let us call it a day. So, I shall see you tomorrow with another topic, which is partial differentiation. Thank you.