In the early 1870s, three economists, William Stanley Jevons, Carl Meneger and Leon Walrus, made similar, though separate, observations in three different countries: England, Austria and Switzerland. They broke with classical economics in terms of the basic goods and services valuation principles. The classical economists based the value of a commodity on its cost of production, thus conferring on it a sort of intrinsic value, likewise from the point of view of the consumer, the value depended on the amount of commodity need or pleasure obtained from its consumption. Including the marginalists, as they came to be called, or neoclassical economists. The utility of a product or service is dependent on the cost of generating the last unit produced or the benefit of the last unit consumed, hence the marginal cost and marginal Benefit ideas. This way of thinking explained why a good for which there was no demand would be disposed of irrespective of its cast production and why air which is so vital to our survival is free. But the diamonds, without which we can exist, are too costly. The core principle of neoclassical economics was to optimize consumer utility and producer income. At the same time marginal costs and marginal utility had a ready-made equivalent in the form of a derivative calculus. A special case of optimization was to optimize utility or minimize costs. Therefore, the use of mathematics has increased in the economics. In 1892 Irving Fisher published his dissertation which used the mathematics extensively. Many economists used calculation tools and tackled various economic issues. It was Paul Samuelson who unified introduced the marginal in his foundations of economic analysis (1947). Today, marginal analysis and calculus usage are permeating every corner of the economy. Differentiation’s importance is not confined to microeconomics. Differentiation is the basis of the dynamic optimization and differentiation equation, which are the key methods of macroeconomics for dynamic analysis. Ultimately, in statistical and econometric estimation and inference, derivative and significant function.

The history of derivatives goes back to the origins of trade in Mesopotamia during the fourth millennium BC. Following the fall of the Roman Empire, contracts for the future production of goods continued to be used in the Byzantine Empire in the eastern Mediterranean and survived in Western Europe under canon law. Capital markets in Italy and Low Countries became more sophisticated during the Renaissance. For the first time in Antwerp, and then in Amsterdam in the sixteenth century, contracts for the future delivery of securities were used on a large scale. By the end of the seventeenth century, derivative trading on securities spread from Amsterdam to England and France. Financial practitioners created graphical devices to represent derivative contracts around 1870. Profit charts made derivatives available to young scientists, including Louis Bachelier and Vinzenz Bronzin, who had the technical skills required for rigorous derivative pricing analysis.

## Marginal Analysis

To define marginal analysis is the evaluation of the costs and benefits of other activities outweighs costs, for example, if a business considers raising the amount of goods it produces, it would carry out a marginal analysis to ensure that the costs of manufacturing more items outweigh the potential costs that follow the decision. Examples include increased labor costs or additional materials you may need to manufacture the goods. Marginal analyzes are helpful in helping individuals and organizations determine how to distribute capital to optimize productivity and benefits and minimize objectives.

In economics, marginal analysis implies we look at the level of consumption or expense. it provides a clear image of the overall cost. For example, the net cost of running a plane from London to New York would be several thousand pounds. But, the cost of transporting an additional passenger is very small for a 50 per cent full aircraft. Therefore, the marginal cost of transporting 102nd passengers is also very low as compared to the overall cost.

Marginal cost (MC) is the cost of generating the final unit of product (the difference between Qn and Qn-1 in gross cost ‘TC’) since MC=∆TC/∆Q.

In business and economics, one important application of mathematics is marginal analysis. The word ‘marginal’ in economics refers to the rate of change, which is to a derivative. Therefore, if C(X) is the total cost of manufacturing X times, then C(X) is referred to as the marginal cost and reflects the instantaneous rate of change in the total cost proportional to the amount of products produced. Therefore, the marginal revenue is the product of the overall income function and the gross contribution function.

### Marginal Benefit Vs Marginal Cost

The marginal benefit is the difference you’ll get when you make up a different choice. In business, this is usually the extra income that the company earns as it rises in production and/or selling additional products. Marginal cost is the additional cost you incur when manufacturing additional product units. Usually, marginal costs decrease if a business delivers a larger number of goods.

### Marginal Analysis and Opportunity Cost

To consider the risk and profit of such practices, you do need to consider the risk of opportunity. Cost of opportunity is the valuable benefit you miss when choosing one option over another. For example, if an organization has place for another worker in its budget and is considering recruiting another worker to work in a warehouse, then a marginal analysis indicates that recruiting another worker has a net marginal

### Profit

In other words, the capacity to produce more products outweighs the labor cost increase. However, hiring that person may not yet be the company’s best decision. For example, if the company knows that recruiting an extra sales agent will have a better net marginal profit, then the correct decision is to employ someone else for distribution rather than someone else to work at the plant. The extra productivity the company should have gained by recruiting someone to work in the warehouse is the cost of opportunity.

### Marginal analysis and Observed Change

In certain cases, making minor organizational changes that make sense for an organization, and then doing a marginal review afterwards to identify improvements in costs and benefits that resulted as a result of such improvements. A business that makes children’s toys, for example, may prefer to raise production by 1 percent to see if improvements are happening in demand and how they affect capital. If the managers find that the advantages of an increase in output outweigh the increased costs incurred by the company, they may opt to retain the higher rate of production or even boost production again by 1 percent to monitor the improvements taking place. Companies may define optimum production levels through minor adjustments and observable improvements.

### Marginal Analysis and Variables

When using marginal analysis for decision-making, you need to take into account cost and production variables. The amounts of the products you produce is evaluated by the most frequent variable companies. Others, however, such as shipping fees, are increasing as you produce and distribute a higher number or weight of products. You can select from a variety of production levels with differing degrees of productivity by making small improvements in output and tracking the gains and costs that follow those improvements.

## Derivative Analysis

Derivative analysis is a powerful diagnostic tool which improves data understanding from pumping tests. The derivative used to measure the pumping test is given by the drawdown data plotted on a graph with semi log axes (linear drawdown and logarithmic time) throughout real-life; we make use of derivatives to measure market income and loss by using illustrations. They also use them to calculate the velocity or distance that they travel in miles per hour or kilometers an hour, and more. Similarly, they are also used to derive calculations of Physics. Differentiation is the method by which to find a derivative. A function’s derivative represents the exit value transition rate with respect to its input value. In contrast, differential means the actual function change.

We are going to show the relation between the marginal concepts and derivative of a function in an economics as marginal cost, marginal utility, and marginal revenue through an exampleю Let’s say the company’s cost function is illustrated as:

C=C (Q) =a+bQ

Where: (C) is the total cost, (Q) is the Output, and (a) and (b) are constants. If the company produces 350 units of output, then the total cost will be:

C1=a+350b

If the company exceeds its output to 40, then the cost will be:

C2=a+400b

The extra 50 units are:

∆C=C2-C1=a+400b-a+350b=50b

Therefore, the total additional cost is 50b and since we have extra 50 units of output, b will be illustrated as the additional cost of each unit.

∆C/∆Q=(C2-C1)/(Q2-Q1)=50b/50=b

The ratio of the additional cost to the additional output is therefore constant, making life easier because whether we add 50, 100 or 1 or even one-tenth of the output unit, the ratio of the additional cost to the additional output will be constant and equivalent to b. let’s suppose the cost function is:

C=C(Q)=a+bQ+cQ^2

So, if we move from 350b units to 400b units, the ratio of additional costs to additional output will be:

∆C/∆Q=(a+400b+160,000c-a-350b-122500c)/(400-350)=(50b+37500c)/50=b+75c

This ratio is not constant, and depends on the additional units produced. But if the outout increased from 350 to 36, and then we will get:

∆C/∆Q=b+710c

The concept of marginal cost is to calculate the extra infinitesimal cost. Let us consider the general case of enhanced production and see how this works, from Q to Q+∆Q we get:

∆C/∆Q=(a+b(Q+∆Q)^2-a-bQ-cQ^2)/∆Q

(b∆Q+2cQ∆Q+c(∆Q)^2)/∆Q

The principle of the marginal cost is to make ∆Q as small as possible, to let it go to zero and determine the factor afterwards. You could say at this point, something divided by zero equal infinity. So, to ease the ratio, we are getting:

∆C/∆Q=b+2cQ+c∆Q

Now if we let delta Q approach zero, the marginal cost of this formula is:

MC=b+2cQ

Make sure that the marginal cost (MC) relies on the output level, so as quickly as we recognize the output level, we can evaluate an infinitesimal quantity of the additional cost of the increased output. The limit of the ratio ∆C⁄∆Q, when ∆Q reaches zero is the derivative of the cost function with respect to output.

## Conclusion

Marginal and derivative analysis are very important in our everyday life especially in business and economics, as it gives a big support for the companies to know their current state to determine whether there is a profit and loss and make them able to predict their state in the future. So, marginal and derivative analysis should be put into more consideration to help stating the financial states in companies and in the markets.