Partial Derivatives

Just as derivatives can be used to explore the properties of functions of 1 variable, so also derivatives can be used to explore functions of 2 variables. In this section, we begin that exploration by introducing the concept of a partial derivative of a function of 2 variables. In particular, we de…ne the partial derivative of f (x; y) with respect to x to be fx (x; y) = lim h!0 f (x+ h; y) f (x; y) h when the limit exists. That is, we compute the derivative of f (x; y) as if x is the variable and all other variables are held constant. To facilitate the computation of partial derivatives, we de…ne the operator