Partial derivatives are used when there are more than two variables under consideration. Say, for example, a three dimensional problem where there is x, y, and z. A partial derivative is basically used to hold all variables constant except for two. The two under consideration can be any two variables desired. In this way, analysis can be simplified.
Put some 3d graphs in here using MatLAB with cut sections.
For three random variables: x, y, and z, the following nomenclature may be used:
which reads: the partial of y with respect to x while holding z constant.