This is primarily a question of understanding the syntax of some output although there might be a bug hidden underneath. Consider the following code:
sage: var('x,t')
sage: q = function('q',x,t)
sage: f = q*q.conjugate()
sage: print f.derivative(x,1)
q(x, t)*D[0](conjugate)(q(x, t))*D[0](q)(x, t) + conjugate(q(x,t))*D[0](q)(x, t)
The answer is supposed to be $d/dx(q\bar{q}) = q_x \bar{q} + q \bar{q}_x$. The second term in the Sage output is correct but I'm having trouble deciphering the first term. Any thoughts?
I think I can narrow down the differences even further. Check it out:
sage: print q.conjugate().derivative(x,1)
D[0](conjugate)(q(x, t))*D[0](q)(x, t)
sage print q.derivative(x,1).conjugate()
conjugate(D[0](q)(x, t))
The independence of order isn't the issue: $q = u + iv$ means that $q_x = u_x + iv_x$, $\bar{q} = u - iv$. So $\bar{q_x} = u_x - iv_x$ and $(\bar{q})_x = (u - iv)_x = u_x - iv_x$.
