First, maple sees the symbol u as a name, not as a function. If u is to be a function of x, y, and z, then you need to enter it as u(x,y,z) to obtain the functional dependence you claim.
Then, the name on the left, u__0, which will look like a subscripted u, needs to be followed by an assignment operator, not merely an = sign. The = sign is used to form equations, Use := (colon equal) as the assignment operator.
Third, if you can apply diff to u__0, but display only an = sign in your post, but yet claim that Maple produced derivatives of some of the terms on the right, then you did more in your worksheet than you reported here. If, indeed, you did what you reported here, then applying diff to u__0 would result in zero because u__0 would not stand for the expression on the right of your = sign. It wouldn't be assigned anything.
In the future, it would be better to upload the whole worksheet via the green arrow in the toolbar at the top of the posting window. Then, we could see exactly what it is that you did, and not have to guess.