Example 4

Given y = ½x, z = x³, with x as the independent variable, then

Δy = ½(x + Δx) - ½x = ½ Δx,
Δz = 3x² Δx + 3x(Δx)² + (Δx)³,
dy = ½ dx, dz = 3x² dx.

The meaning of the symbols for increment and differential in this example will be different if we take y as the independent variable. Then x and z are functions of y.

x = 2y, z = 8y³.

Now Δy = dy is just an independent variable, while

Δx = 2(y + Δy) - 2y = 2 Δy,

Δz = 8(y + Δy)³ - 8y³=
= 8[3y² Δy + 3y(Δy)² + (Δy)³]=
= 24y²Δy + 24y(Δy)² + 8(Δy)³.

Moreover,

dx = 2 dy, dz = 24y² dy.

We may also apply the differential notation to terms. If τ(x) is a term with the variable x, then τ(x) determines a function f,

τ(x) = f(x).

and the differential d(τ(x)) has the meaning

d(τ(x))=f'(x)dx.