You might find the recent exchange about differentiation on the Maxima mailing list interesting. I believe that this definition extension made it through all of the Maxima test cases and has been incorporated into the most recent version of Maxima. At 11:12 AM 2/16/2012, Stavros Macrakis wrote:

Actually, a little reflection makes it clear that diff(f(x),g(x)) should not be an error or a noun form, because in fact it is perfectly well-defined and meaningful:

diff(A,B) == ev(diff(A)/diff(B),del)

For example,

diff(sin(x),x) == cos(x)*del(x)/del(x) == cos(x) << diff(...,V) reduces to normal differentiation diff(sin(x),cos(x)) == cos(x)*del(x)/(-sin(x)*del(x)) == -cos(x)/sin(x) << in diff(f(x),g(x)), del(x)/del(x) => 1 diff(x*y,x^2+y^2) == (x*del(y)+y*del(x)) / (2*y*del(y) + 2*x*del(x)) << consistent handling of multiple-variable case

How useful is this? I'll leave that to others....

Right now, Maxima doesn't do much of interest with del(...) -- not even giving an error for del(1) -- but it does do ev(del(x^2),del) == diff(x^2) == 2*x*del(x). You can even try ratsubst('diff(x,y),del(x)/del(y),...) on del expressions and get things that look reasonable (and useful?)....

One problem: diff(del(x)) => 'diff(del(x),x)*del(x), which is I suppose correct, but ugly and hard to work with. Can be deferred for now.

Any objections to this definition of diff with a second argument which is an expression?

-s