Partial Derivatives of Functions of More than one independent variable

 3 Now consider a function,  f(x,y), which has two independent variables   x  and  y. One can extend the definition of the limit for a function of two independent variables in a similar fashion leading to the partial derivatives.  (Nonrigorous)   So the partial derivative of f(x,y) with respect to x   (holding  y  constant)  is:                                         fx(x,y)  =   lim [ f(x + h, y)  -   f(x,y)]/ h                                                      h à 0 Likewise the partial derivative of f(x,y) with respect to y  (holding x constant) is:                                         fy(x,y)  =   lim [ f(x, y + k)  -   f(x,y)]/ k                                                      k à 0   With a function,  f(x,y), of two variables   fx(x,y)    represents the slope of  f(x,y) in the x-direction whereas  fy(x,y)  represents the slope of  f(x,y)  in the y-direction.     Notation:   fx(x,y)    =  ∂f(x,y)/ ∂x   and    fy(x,y)    =  ∂f(x,y)/ ∂y       Click here to examine the physical interpretation of the partial derivative. 4 Click here for two examples.