Differentiation of Functions with two or more than two Independent Variables (Partial Derivatives)

Partial Derivatives:

In economics relations contain two or more than two independent variables about whose use economists and managers of business firms have to take decisions.

For example, demands for the product of a firm depends on its price, income of the consumers, price of its substitute, advertising outlay made by the firm to promote the sales of its product and some others.

Further, output of a product depends on the amounts of labour, capital, raw materials etc. used for the production of a commodity. Other examples of functions from economics and business with two or more independent variables can be given.

When a function has two or more independent variables and each of them has an effect on the value of the dependent variable, we use the concept of a partial derivative. It is called partial derivative because in this the effect of only a part of influences on the dependent variable is examined. A partial derivative of a function measures the marginal effect of a change in one variable on the value of the dependent variable, holding constant all other variables.

Thus in a function, Y = f(x₁, x₂, x₃), partial derivative of y with respect to x₁, will show the marginal effect of a very small change in x₁, keeping constant x₂, x₃. By convention and to distinguish it from derivative of a function with one independent variable, for partial derivative we use lower case delta (∂) instead of lower case d. However, rules of differentiation in finding partial derivatives are the sameas explained above in case of derivative of a function with a single independent variable.

It is worth nothing that in multivariable function, the partial derivative of one independent variable depends on the values at which other independent variables are held constant. That is why the expression for partial derivative of profit function of a firm with two independent variables, products x and y, indicates that ∂π/∂x depends on the level at which the variable y is held constant.

Similarly, the partial derivative of profit function with respect to y indicates that it depends on the value of the variable x which is held constant. The economic reasoning for this will become clear if we take a two factor production function q = f (L, K). In this partial derivative of production function with respect to labour (L), that is, ∂q/∂L implies marginal product of labour.

Now, as is well known, marginal product of labour depends not only on its own skill and efficiency, but also on with how much capital (K) he has to work with. Generally, the greater the amount of capital, the higher will be marginal productivity of labour, the other things remaining the same.

To illustrate the concept of partial derivative we take the example of profit function with sales of two products as independent variables:

π = = f(x, y) = 50x – 3x² – xy – 4y² + 60y

Where π represents profits, x and y are the sales of the two products being produced by a firm. The function represents that profits of a firm depend on the sales of two products produced by it.

Determining the partial derivative of profit (k) function with respect to sales of the product X treating sales of y as constant from the above profit function we obtain

∂π/∂x = 50 – 6x – y

Thus, with partial derivative we are able to isolate the marginal effect on profit (π) of the change in the sales of the product x, keeping the sales of products y as constant.

Note that in finding partial derivative of the profit function with respect to x, fourth and fifth terms in the profit function are not considered because they do not contain the variable x.

Likewise through partial derivative we can separate the marginal effect of the variation in sales of the product y on profit (π) while holding x constant.

Thus, partial derivative of profit (π) function with respect to y is

∂π/∂y = -x- 8y + 60.