Command line (6)

# Define a "new" function which contains the
# derivative of the original function
#
  g:= (x, y)-> D(f)(x)*y;

g defined as (x,y) arrow D(f)(x)*y;

Like i mentioned earlier this is more than likely down to my previous education but noone can be expected to make sense of this notation, and even more importantly, figure out how to reproduce it in Maple. I've seen your document but i cannot recreate it.
 

@Scot Gould Thanks for the detailed answer! 

Of course this was just a simple example of the problem, and to restate my issue:

what i often require is a way to define a new function which has the derivative of a previously defined fucntion as a part of it, say

f(x,y) = d/dx g(x) * y

and then later evaluate f(2,3) or so, expecting the d/dx to act as an operator rather than keeping "x" as a variable, and trying to evaluate f(2,3) = d/d2 g(2)*3.

This may simply be a fault of my previous experience with math programs like Maple, especially since im not used to seeing the 

g:= x->( d/dx f(x) )

notation.

But intuitively, f(x) is assigned a "static" function first calculated​ as the derivative of g, then assigned to f as a string, not related to the d/dx operator any longer.

This of course breaks the functionality of S(i,j) = d/du_i (expression) + d/du_j(other expression).

>>I couldn't help but notice that the document you provided is written in "worksheet mode" as opposed to document mode. Does this matter for the demonstration, or do i simply lack the knowledge of how to the |_x=x1 operation?