Partial Derivatives

11.3 PARTIAL DERIVATIVES

Partial derivatives are used to study the rates of change of functions of two or more variables. In general the rate of change of z = f(x, y) will depend both on the rate of change of x and the rate of change of y. Partial derivatives deal with the simplest case, where only one of the independent variables is changing and the other is held constant.

Given a function z = f(x, y), if we hold y fixed at some constant value b we obtain a function

g(x) = f(x,b)

of x only. Geometrically the curve z = g(x) is the intersection of the surface z = f(x, y) with the vertical plane y = b. The rate of change of z with respect to x with y held constant is the slope of the curve z = g(x). This slope is called the partial derivative of f(x, y) with respect to x (Figure 11.3.1(a)). There is also a partial derivative with respect to y (Figure 11.3. 1(b)).

Figure 11.3.1(a): Partial Derivatives

Figure 11.3.1 (b)