Total differential

In the case of a function of a single variable the differential of the function y = f(x) is the quantity dy = f '(x) Δx.

This quantity is used to compute the approximate change in the value of f(x) due to a change Δx in x. As is shown in Fig. 2,

Fig 2.

Δy = CB = f(x + Δx) - f(x), while dy = CT = f '(x)Δx.

When Δx is small the approximation is close. Line AT represents the tangent to the curve at point A. In the case of a function of two variables the situation is analogous. Let us start at point A(x₁, y₁, z₁) on the surface z = f(x, y) shown in Fig. 3 and let x and y change by small amounts Δx and Δy, respectively. The change produced in the value of the function z is

Δz = CB = f(x₁ + Δx, y₁ + Δy) - f(x₁, y₁).

An approximation to Δz is given by .

When Δx and Δy are small the approximation is close. Point T lies in that plane tangent to the surface at point A.

The quantity .

is called the total differential of the function z = f(x, y). Because is customary to denote increments Δx and Δy by dx and dy, the total differential of a function z = f(x, y) is defined as

.

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