Example 2

Let y = x³, so that y' = 3x². According to the Increment Theorem,

Δy = 3x² Δx + ε Δx

for some infinitesimal ε. Find ε in terms of x and Δx when Δx ≠ 0. We have

We must still eliminate Δy. From Example 1 in Section 2.1,

Substituting, 

ε = (3x² + 3x Δx + (Δx)²) - 3x².

Since 3x² cancels,

ε = 3x Δx + (Δx)².

We shall now introduce a new dependent variable dy, called the differential of y, with the equation

dy=f'(x)Δx.

dy is equal to the change in y along the tangent line as x changes to x + Δx. In Figure 2.2.3 we see dy and Δy under the microscope.

To keep our notation uniform we also introduce the symbol dx as another name for Δx. For an independent variable x. Δx and dx are the same, but for a dependent variable y, Δy and dy are different.