2.6.2. Partial derivatives

Let be defined on area D. We assume that the argument y is constant and consider the resulting function in one variable x . Give the variable x an increment Δx . Incrementing Δx will cause the function to increment . The final limit of the ratio of the increment of the function Δz x to the increment of the argument Δx at is called a partial derivative of the first order in x and is denoted by i.e. .

If we consider the argument x constant and consider the function as a function of one variable y , the increment Δy will cause an increment of the function . The final limit of the ratio of the increment of the function Δz y to the increment of the argument Δy at is called a partial derivative of the first order in y and is denoted by i.e. .

The symbols are also used to denote partial derivatives: .

Partial derivatives of the second order of the function are called partial derivatives of its partial derivatives of the first order: .

And if derivatives are continuous.

Derivatives of higher orders are calculated similarly.