Notion of differential of a function of two variables

 Notions of differentials

In linear approximation and differentials for a function of one variable, the differential of y written as dy is defined as f'(x)dx. The differential is used to approximate Δy = f(x + Δx) - f(x) where  Δx = dx. Extending this notion to the linear approximation of a function of two variables at the point (x₀, y₀)  leads to the concept of total differential of a function of two variables.

Definition

Let z = f(x,y) be a function of two variables with (x₀, y₀) and (x₀ + Δx , y₀ + Δy) both in the domain of f. If f is differentiable at the point (x₀, y₀) , then the differentials for dy and dx are written as dx = Δx and dy = Δy.

The differential dz called total differential of z = f(x,y) is given by:

dz = fₓ(x₀, y₀)dx + f y(x₀, y₀)dy.

This formula can be used to approximate Δz as shown in the following example:

Example

Find the differential dz of the function f(x, y) = 3x ²- 2xy + y² and use it to approximate Δz at the point (2, -3). Use Δx = 0.1 and Δy = -0.05. What is the exact value of Δz ?

Solution

Let's use the formula:  dz = fₓ(x₀, y₀)dx + f y(x₀, y₀)dy

Let's substitute  (x₀, y₀) :

 dz = fₓ(2, -3)dx + f y(2, -3)dy

Let's calculate fₓ(2, -3 and  f y(2, -3:

fₓ (x, y) = 6x -2y  

fₓ(2, -3 = 6(2) - 2(-3) = 12 + 6 = 18

fy(x, y) = -2x + 2y

 f y(2, -3) = -2(2) + 2(-3) =  -4 - 6 = -10.

Let's substitute  fₓ(2, -3) dx, f y(2, -3) dy in the formula of the total differential:

dz = 18(0.1) - 10(-0.05) = 2.3

The exact value of Δz is given by:

Practice

Find the differential of the function:

and use it to approximate Δz au point (1, -1). Use   Δx = 0.03 and Δy = -0.02. What is the exact value of Δz ?