I have been discussing building a calculator that will be infinitely superior in quality (both user quality and mathematical capability) to Mathematica with one of my good friend and archnemesis who is using Scheme (the dialect of lisp).

I was thinking about how easy that'd be with the "Lambda" form. Basically the lambda form is defined as follows
lambda (x) (+ x 4)  //a function of x such that it equals x+4
lambda (<formal-parameters>) (<body>) /*the formal parameters akin to java's public static void(<formal-parameters>) and the (<body>) akin to java's, well, body
I have seen it make derivatives infinitely easier, not to mention how easy functions would become. But it seems rather abstract: making a public static void method to make public static void methods within a method.

That stumped me. And unfortunately, my good friend and archenemy has all ready done calculus! (Horrors!) So time is of the essence.