Application of Derivatives

**Cost Function**: If

*C*is the total cost incurred in producing and marketing

*x*units of a certain product, then a function relating to

*C*and

*x*is called a cost function and is generally depicted as

*C*=

*C*(

*x*) in the explicit form.

In the implicit form, the cost function can be written as

*$f\left(C,x\right)=0$*

If

*C*is the cost incurred in producing

*x*units of a commodity, then

*$C=2+5x,C=3+2x+5{x}^{2},C=a{x}^{2}+bx+c$ etc are the cost functions.*

The total cost of producing

*x*units of a commodity consists of two parts:

(i) Fixed Cost

(ii) Variable Cost

(ii) Variable Cost

**Fixed Cost:**It is the sum of all costs that are independent of the level of production.

Fixed cost remains constant at all levels of production and it generally includes rent, depreciation, insurance and interest etc.

**Variable Cost:**It is the sum of all costs that are dependent on the level of production.

Variable cost varies as the level of production varies. It includes the cost of labour, material, advertising expenses, etc.

Clearly, Total cost = Fixed cost + Variable Cost.

The graph of the function $C=C\left(x\right)$ is called the cost curve. It is evident from the cost curve the fixed cost is determined by the intersection of the cost curve with the cost axis.

**Demand Function:**The quantity demanded of a given commodity depends upon several variables such as the price of the commodity, prices of the substitutes and complements, disposable income, wealth, tastes, habits, etc. However, in elementary economics analysis, demand is considered dependent on the commodity price. We will assume that the quantity demanded depends on the price only.

If

*p*is the price per unit of a certain commodity, then the demand function in explicit form is written as $x=f\left(x\right)$.

In implicit form, the demand function is written as $F(x,p)=0$.

In the demand, function $x=f\left(p\right),x$ is the dependent variable and

*p*is the independent variable. Also, $x\ge 0$ and $p\ge 0$, because negative quantities and negative prices are meaningless.

Usually, as the price increases, demand decreases and due to a decrease in price demand increases. Therefore, the demand curve is always decreasing as shown below:

**Supply Function:**A relatio…

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