Example 2 (Continued)

The function z = x² - 3xy³ is smooth for all (x, y). Express Δz in the form

Δz = dz + ε₁ Δx + ε₂ Δy

at an arbitrary point (x, y) and at the point (5,4). We have

Δz = (2x - 3y²) Δx - 6xy Δy + Δx² - 3x Δy² - 6y Δx Δy - 3 Δx Δy²,
dz = (2x - 3y²) Δx - 6x3' Δy.

Then

Δz = dz + Δx² - 3x Δy² - 6y Δx Δy - 3 Δx Δy².

Each term after the dz has either a Δx or a Δy or both. Factor Δx from all the terms where Δx appears and Δy from the remaining terms.

Δz = dz + (Δx - 6y Δy - 3 Δy²) Δx + (- 3x Δy) Δy.

Then

Δz = dz + ε₁ Δx + ε₂ Δy,

where

ε₁ = Δx - 6y Δy - 3 Δy², ε₂ = - 3x Δy.

At the point (5, 4),

Δz = dz + ε₁ Δx + ε₂ Δy,

where

ε₁ = Δx - 24 Δy - 3 Δy², ε₂ = -15 Δy.