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Total Differential

It my be recalled that if y = f (x) is a differentiable function of x, then the derivative, dy/dx, of y with respect to x is given by

dy/dx = lim  Δy/Δx,

where Δ y denotes the increment in y due to an arbitrary small increment Δ x in the variable x. Hence

Δy/Δx = dy/dx             approximately when Δ x is small.

i.e,     Δy = dy/dx Δ x  approximately when Δ x is small

In other words, if x changes by Δx, then the change in y, Δy, is approximately dy/dx times the change in x. Thus the expression

Δy = dy/dx Δ x

gives the approximate increment in y for an arbitrary small increment Δx in x.

We now generalize this notion t the case of a function z = f (x,y) of two variables. The partial derivative, ∂z/a∂x, gives the rate at which z changes when x changes (y remaining constant). Hence, if Δ x z denotes the increment in z due to an arbitrary small  increment Δ x in the variable x from a point (x,y), then

Δx z = ∂z/∂x Δx  approximately.

In the same way, if Δyz denotes the increment in z due to an arbitrary small increment Δ y in the variable y form the point (x,y), then

Δy z = ∂z/∂y Δy  approximately.

Thus the increment in z when x varies alone is represented approximately by ∂z/∂x Δ x and the increment when y varies alone approximately by ∂z/∂y Δ y. There remains the important problem of expressing the increment in z when x and y vary together. It has been established ( under some continuity conditions) that the increment in the function z = f (x,y) corresponding to arbitrary small increment  Δ x and Δ y in x and y is approximately

Δz = ∂z/∂x Δ x + ∂z/∂y Δ y.

This is called the differential of z and is denoted by dz. For convenience we denote the arbitrary increment Δ x by dx and call it the differential of the independent variable. Similarly, Δy is written by and called the differential of y.The differential of the dependent variable z is then defined in terms of the independent differentials by the formula

dz = ∂z/∂x dx + ∂z/∂y dy.

It must always be remembered, in using this formula, that dx and dy are no more than arbitrary increments in the independent variables. An alternative notation for the differential of z is

df = fx dx + fy dy

The expression dz = ∂z/∂x dx + ∂z/∂y dy is often called the total or complete differential of the function z = f (x,y).

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