If I evaluate D[f[1, 1], x] the output is zero, as it should, because f[1,1] does not depend on x. The D operator acts on expressions that contain variables with symbolic names such as x,y etc. With D you take the derivative of the expression with respect to a variable.

The Derivative operator acts on functions, which are constructs that have numbered slots: slot number 1, slot number 2 etc. With Derivative you take the derivative with respect to a slot, which we identify with its number, not with a variable name. The function f[x_,y_]=x^2y has two slots, forget about the variables which are used to define f. Then Derivative[1,0][f][x,y] is the first derivative with respect to the first slot and zeroeth derivative with respect to the second slot. Then the slots are filled in with whatever is x and y.

When we call Derivative[0,1][f][whatever arguments] the f object must be something for which f[x,y] makes sense. It does not make sense to call Derivative[0,1][x^2y], because (x^2y)[3,2] has no meaning.

What you call Subscript[f, yx] would correspond to Derivative[1,1][f][whatever arguments].

Thanks so much for your reply. I was under the impression that D was a shortcut for Derivative, which is apparently not the case. However there seems to still be a problem with D[f[1, 1], x] and D[f[1, 1], y], since it still gave me an answer without any warnings or errors. Additionally Derivative seems a little odd since all of its trailing terms are not enclosed in a single set of [ ] like the rest of Mathematica's commands. Could you explain why Mathematica chose to handle Derivative differently.

I found your response very helpful. However your response brought up another question. Whenever you are taking the 2nd derivative of a function with 2 variables, you actually end up with 4 solutions: Subscript[f, xx], Subscript[f, yy], Subscript[f, xy], Subscript[f, yx]. Taking the 2nd derivative of xx and yy is not a problem for the Derivative command. However, is there a way to calculate the 2nd derivative of first taking the derivative of x and then taking the derivative of y and subsequently the y and x variables using the Derivative command? I certainly can calculate the derivatives manually but with the ability to select the derivative level with the variable using the Derivative command, there might be a better way to perform the calculations within the command.

Thanks again,

Mitch Sandlin