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Functions and Variables and their Differentiation

The article " On Partial Differentiation " by M. Bruckheimer and L. E. Scraton {Gazette, October 1969) raises once again the question of the superiority, or otherwise, of the Leibnitz notation dyjdx for a derivative vis-a-vis the Lagrange form/'(#). Almost a decade ago I circulated a note amongst my colleagues which attempted to clarify the role most appropriate to each form, but in the intervening years I have not seen the distinction clearly drawn in any of the many published discussions of this topic. To begin with the notion of a function: I find it useful to introduce students in the first place to the idea of a " black box " with one input and one output. They soon accept the fact that the mechanism of such a box can operate in the most bizarre way, thus overcoming the usual objection that a function must be a formula of some kind. This concept is then developed mathematically in two different, but complementary, ways. In the first, emphasis is placed on the various black boxes themselves. Each is given a characteristic symbol, e.g. <f> for the black box in Figure 1, and is now called a