The literature on gradient and differential microphone arrays makes a distinction between the two types, and nevertheless shows how both types can be used to obtain the same directional responses. A more theoretically sound rationale for using delays in differential microphone arrays has not yet been given. This paper presents a gradient analysis of the sound field viewed as a spatio-temporal phenomenon, and gives a theoretical interpretation of the working principles of gradient and differential microphone arrays. It shows that both types of microphone arrays can be viewed as devices for approximately measuring spatio-temporal derivatives of the sound field. Furthermore, it also motivates the design of high-order differential microphone arrays using the aforementioned spatio-temporal gradient analysis.