Mathematica has two different ways to make derivatives. The operator D makes derivatives of expressions, and outputs other expressions. The operator Derivative makes derivatives of functions and outputs functions. The syntax of Derivative is like this:

f[x_, y_] := 4 - x^2 - y^2;
Derivative[1, 0][f][x, y]
Derivative[1, 0][f][1, 1]

as you can see, you can calculate the derivative in a given point just by giving the coordinates. For functions of one variable, there is the much more convenient shorthand f'[x], which is equivalent to Derivative[1][f][x]. You can find documentation in the page defining derivatives

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