6.3.3 Derivatives: Examples, Part 3

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Quantitative Foundations for International Business

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University of London

Quantitative Foundations for International Business

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Cours 3 sur 6 dans Specialization International Business Essentials

This course provides the essential mathematics required to succeed in the finance and economics related modules of the Global MBA, including equations, functions, derivatives, and matrices. You can test your understanding with quizzes and worksheets, while more advanced content will be available if you want to push yourself. This course forms part of a specialisation from the University of London designed to help you develop and build the essential business, academic, and cultural skills necessary to succeed in international business, or in further study. If completed successfully, your certificate from this specialisation can also be used as part of the application process for the University of London Global MBA programme, particularly for early career applicants. If you would like more information about the Global MBA, please visit https://mba.london.ac.uk/. This course is endorsed by CMI

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Derivatives

An important topic in many scientific disciplines- including economics- is the study of how quickly quantities change. The concept used to describe the rate of change of a function is known as the derivative. In this lecture, we will define the derivative of a function, and share some of the important rules for calculating it.

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George Kapetanios
Professor of Finance and Econometrics
King's College London

[MUSIC]

So, last time we dealt with derivatives.

I gave you some definitions of what derivatives are and to give you

a bit of content and environment as to where derivative can be important.

Let's have an example.

So this is average revenue function which is

represented by 15 as a value and q as the quantity.

Now you as the producer if for instance you are producing something

would be interested to know what effect the production has on your revenue.

And you have an average revenue.

What does it mean?

It means on an average when you increase your quantity by say, 1 unit,

you decrease your average revenue by 1 unit respectively.

But it is it through for all the units?

Is it true that if you produce say thousands units or

millions units, the effect will be the same.

And this is where marginal revenue or derivatives are quite important.

Let's start first of all

by giving it a content that relates to the two examples that provide you earlier.

Simply saying that one function of a our interest would be linear function and

in other function, our interest will be quadratic function.

So for instance, when we talked about linear functions

Regardless where you take the points, The rate of change will be the same.

So, here or here, it will be the same, and it will correspond to

the gradient of the linear function, which is represented by,

This letter a.

In quadratic function, if you remember, If for

instance, this is a quadratic function, Which

has the following form, the gradient will depend very much,

On your location, On the graph.

So average revenue can be quite deceiving because it depends where you

take this average revenue from.

So to get a better idea of what effects your production has

on your revenue rather than looking at averages which is

basically saying what is the gradient of those lines.

Or those lines which is represented simply by functions similar to that.

We would like to have a look at specific points on the graph.

And not a linear approximation of two points which are near each other.

So, let's try to work out with this average revenue function.

[COUGH] This average function as I said represents a similar

function to let's say, that if we increase production by 1 unit,

our average revenue will decrease by 1 unit respectively.

But whether this is the case for all the output points,

this is the question that we're after right now.

So in order to translate this average into total revenue, what we need to do,

We need to multiply this by the quantity itself.

Because what is essentially average revenue?

Average revenue is total revenue divided by quantity.

Therefore, total revenue = average revenue times the quantity,

which is exactly what I did here.

[COUGH] So

in this case we'll have 15q- q squared.

Now this is the total revenue function.

For this total revenue function, we can find out what will

be the rate of change for each and every production unit.

Or in other words, we call this a marginal revenue function.

Sometimes marginal revenue will be the same value as an average revenue but

as I said before, it's not necessarily so.

So in order to find the margins revenue, what we need to do,

we need to find the derivative of the total revenue function.

Because now you're already in position to know what

were the derivative of linear function and quadratic function,

I can directly go into finding that this will be 15- 2q.

Now as you see this and this function are very similar.

However, there is a big distinction between those two

functions is that the value that stands in front of q,

in front of production is 2 here whereas as it's 1 here.

And what does this function say which this function doesn't?

This function can say to you what will be the effect of

production unit on your revenue,

regardless where you take the production unit from.

Whether you take the production unit from 100 to 101 unit,

or whether you take the production unit from millions and

you increase it by a tiny bit?

So this will be a much more accurate representation

of the fact on your revenue on the rate of

change in revenue, compared to this one.

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