To extend the concept of a derivative to a function of more than one inde-

pendent variable in a manner consistent with an ordinary derivative, the idea of a cutting plane is utilized. Let us consider the case of a function f(x , y) of

the two independent variables x and y, which is defined in a neighbourhood

of the point (x0, y0). Cutting the surface

zf (x, y)

by the plane y/y0 gives a curve of intersection in the plane y/y0 with the equation

zf (x, y0):

The gradient of this curve at the point (x0, y0) is defined to be the first order

partial derivative of f(x , y) with respect to x at (x0, y0). To distinguish this

partial derivative from an ordinary derivative with respect to x , it is denoted

by writing (1f /1x)(x0, y0). Thus, from the definition of an ordinary derivative, as z/f(x , y0) is only a function of x it follows that



lim h00

f (x0 h, y0) f (x0, y0)



provided this limit exists.