There's been a little discussion on the Maxima email list about diff(expression1,expression2), where expression2 isn't an atomic symbol, and what this might mean. For example, we know that diff(x^2,x)=2*x. But what is the meaning of diff(x^2,x*y) ? In Maxima, diff(expression1) (without a 2nd argument), does a "total differential", and leaves del(x), del(y), etc., in the expression. Thus, diff(x*y) = x*del(y)+y*del(x). One possibility for defining diff(expression1,expression2) is diff(expression1,expression2) = diff(expression1)/diff(expression2),factor (we hope that factor will cancel out any del(x), del(y), etc.) Thus, diff(x^2*y^2,x*y) = 2*x*y*(x*del(y)+y*del(x))/(x*del(y)+y*del(x)) = 2*x*y Problem: There is no guarantee that we can get all the "del(variable)" items to cancel, is there ?? Now we have an analogous problem with integrate(expression1,expression2). integrate(x^2,x) = x^3/3 But what is integrate(x*y,x*y)? If we let a new variable z=x*y, then we can do the following: integrate(x*y,x*y) = integrate(z,z) = z^2/2 = (x*y)^2/2 What about integrate(sin(x),x*y) ? integrate(sin(x),x*y) = integrate(sin(x),z) = integrate(sin(z/y),z) = -y*cos(z/y) = -y*cos(x) Is there any standard way to do this? Can we extend this to definite integration: integrate(expression1,expression2,low,high) ?? What is the relationship between diff(expression1,expression2) and integrate(expression1,expression2) ?? Normally, we expect diff(integrate(expression,x),x)=expression, but what if x is no longer a variable, but an expression??