Simple ways to evaluate a derivative at a point?

Question

The contrast in behavior between, say,

f[x_] = Sin[x^2];
f'[2]

vs.

u[x_, y_] = Cos[x + y^2];

has always bothered my students---and me! (Why does it do it this way?)

Anyway, I tell them to handle ${\partial u\over \partial x}\Big|_{x=2}$ via

Is this the "simplest" way? How else might we accomplish this?

I realize that this is a subjective question, but our context is this is a class where we use Mathematica as a tool rather than a class centered on Mathematica itself. Thus, I want to keep the commands and code required as reasonably elementary as possible (and preferably resembling their paper-and-pencil mathematical work).

Answer 1

You could use as input:

 Derivative[1, 0][u][2, y]

You can implement the formating rule (Thanks to @Rojo)

 Format[Derivative[i_, j_][a_]] := Row[{"\[PartialD]"^(i + j), a}]/
   Row[Row[{"\[PartialD]", #}] & /@ {"x"^i, "y"^j}] 
 Format[(h : Derivative[i_, j_][a_])[x0_, y0_]] := 
   RawBoxes@SubscriptBox[RowBox[{MakeBoxes@h, "\[VerticalLine]"}], 
   MakeBoxes@Row[{Row[{x, "=", x0}], ",", Row[{y, "=", y0}]}]]

so that

   Derivative[1, 2][u][2, 3]

returns

Note that I arbitrary assumed the variables were x,y since I have no way to know from the input alone what the variables are.

This solution also works with input such as

Derivative[1, 2][f][1, x]

Derivative[0, 0][f][1, 3]

Derivative[2, 0][f][1, 3] 

More generally, provided you use (from @Jens see link below)

Derivative /:
MakeBoxes[Derivative[\[Alpha]__][f1_][vars__Symbol],
               TraditionalForm] :=
Module[{bb, dd, sp},
            MakeBoxes[dd, _] ^=
            If[Length[{\[Alpha]}] == 1, "\[DifferentialD]", "\[PartialD]"];
            MakeBoxes[sp, _] ^= "\[ThinSpace]";
            bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
            FractionBox[ToBoxes[bb[dd^Plus[\[Alpha]], f1]],
            ToBoxes[Apply[bb,  Riffle[Map[bb[dd, #] &, 
            Select[({vars}^{\[Alpha]}), (# =!= 1 &)]], sp]]]]]


D[u[x,y,z],{x,2},{y,3},{z,2}]

Or extending the above rule to arbitrary dimensions (again from @Rojo I am just a secretary here !)

Format[Derivative[i__][a_]] := 
Row[{"\[PartialD]"^Total[{i}], a}]/ 
Times @@ MapIndexed[Subscript["\[PartialD]x", First@#2]^#1 &, {i}]

and

Format[(h : Derivative[i__][a_])[vals__]] := 
RawBoxes@SubscriptBox[RowBox[{MakeBoxes@h, "\[VerticalLine]"}], 
ToBoxes@Row[ Riffle[MapIndexed[ 
Row[{Subscript["x", First@#2], "=", #1}] &, {vals}], ","]]]

so that

Derivative[2, 3, 1][u][1, 2, 3]

returns

Answer 2

Perhaps you could add an input alias such as

AppendTo[CurrentValue[$FrontEndSession, InputAliases], 
 "der" -> SuperscriptBox["\[SelectionPlaceholder]", 
   TagBox[RowBox[{"(", "\[Placeholder]", ")"}], Derivative]]]

and then use Esc+der+Esc, and enter the derivatives separated by commas