We studied the derivative as the rate of change of functions of the type y = f (x) – that is, functions of one variable only. This section is concerned with the generalization of some of the previous concepts to functions of two variables.

For example, consider a function of two independent variables written in the form z = f (x,y). We might wish to study the rate at which z changes as x changes if y is held constant. By holding y as a constant z = f (x,y) essentially becomes a function of x alone and we can calculate its derivative with respect x. This derivative is called the partial derivative of z (or f ) with respect to x and is denoted by

∂z/∂x, ∂/∂x [f  (x,y)], fx (x,y) or zx (x,y).

Similarly, the partial derivative of z (or f) with respect to y is denoted by

∂z/∂y, ∂/∂y [ f (x,y)], fy (x,y) or zy (x,y)

and gives the rate of change of z with respect to y, with x held constant. In terms of limits, we have the following:

Definition. If z = f (x,y), the partial derivativae of z with respect to x, denoted by ∂z/∂x, is the function given by

∂z/∂x = lim f (x+ h, y)- f (x,y)
          h→0         h

Provided this limit exists.

The partial derivative of z with respect to y, denoted by az/ay, is the function given by

∂z/∂y = lim f (x, y+ k) - f (x,y),
          k→0       k

Provided this limit exists.

From the definition, we see that to find ∂z/∂x, we treaty y as a constant and differentiate z with respect to x in the usual way. Similarly, to find ∂z/∂y we treat x as a constant and differentiate z with respect to y.

Partial derivative of z = f (x,y) with respect to x evaluated at the point (x₀,y₀) is denoted by

(∂z/∂x) ₍x₀, y₀₎ , fx (x₀, y₀)   or zx (x₀, y₀),

Similarly, partial derivative of z = f (x,y) with respect to y evaluated at (x₀,y₀) is denoted by

(∂z/∂y) ₍x_0, y₀₎ , fy (x₀, y₀)   or zy (x₀, y₀),

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