Differential Calculus of Functions of Several Variables

The differential calculus of functions of one variable made it possible to understand the relationship of how the change in the independent variable x affects the change in the dependent variable y, where y = f ( x ) $$y=f(x)$$ . However, in economic problems, the dependent variable y is most often a function of several independent variables x 1 , x 2 … , x n $$x_{1}, x_{2}\ldots ,x_{n}$$ , i.e. y = f ( x 1 , x 2 … , x n ) $$y=f(x_{1}, x_{2}\ldots ,x_{n})$$ . Therefore, in this chapter we develop differential calculus for functions of several variables and thus deepen our understanding of the interrelationship among these variables. This will enable us to establish how a change in one independent variable affects a change in the dependent variable, assuming that the remaining variables remain the same. A function of n variables has n partial derivatives, which can be written using a vector we call the gradient, while the second-order partial derivatives can be written using a symmetric Hessian matrix. We continue by presenting the chain rule for functions of several variables, which enable us to find derivatives of composite functions, and then move on to the first order total differential and the equation of the tangent plane to a surface. Finally, by using the first order total differential, we derive the Implicit function theorem, which gives us the rule to differentiate implicitly defined functions.