Increment and Total Differentiatial

We now introduce two new dependent variables, the increment Δz and the total differential dz.

The increment Δz depends on the four independent variables x, y, Δx, Δy, and is equal to the change in z as x changes by Δx and y changes by Δy. Thus

Δz = Δf(x, y, Δx, Δy),

where Δf is the function

Δf(x, y, Δx, Δy) = f(x + Δx, y + Δy) - f(x, y).

When x and y are independent variables, dx and dy are the same as Δx and Δy. The total differential dz depends on the four independent variables x, y, dx, and dy. Thus

dz = df(x, y, dx, dy),

where df is the function

df(x, y, dx, dy) = f_x(x, y) dx + f_y(x, y) dy.

Figure 11.4.1 shows Δz under the microscope.

Figure 11.4.1

Example 1: Increment and Total Differential (Product Function)

Example 2: Increment and Total Differential (Biquadratic Function)