# What is the mathematical meaning behind D[f]?

y = a + b x;


I can understand this output of the ordinary differentiation of y w.r.t. x

D[ y, x]

b


but I don't understand the mathematical meaning of this output

D[y]

a + b x


In fact there is no matching usage of D[f] (with only single argument) in the Mathematica documentation. Usually Mathematica will flag up argument count mismatch when running, e.g.

"... called with m argument; n arguments are expected"


Can anyone help me to understand the meaning of D[f]? Is this a bug or undocumented behavior in Mathematica?

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It's not a bug if you consider this behavior as a logical continuation of the following permissible syntax:

D[a + b x^3, x, x, x]

(* ==> 6 b *)

D[a + b x^3, x, x]

(* ==> 6 b x *)

D[a + b x^3, x]

(* ==> 3 b x^2 *)

D[a + b x^3]

(* ==> a + b x^3 *)


The point is that a Sequence of variables is allowed following the first argument of D. And the empty sequence in the last example is a special case of this.

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That's a very good and compelling reasoning for the output it provides... In my earlier comment, I didn't say undefined syntax, but invalid syntax (I can see how "invalid" is incorrect terminology as well... I should've said "undocumented"). I would consider this to either be a documentation oversight (if this use is intended) or speculative... the definition might as well be D[f_] := f as I had said in my comment :) – R. M. Apr 1 '14 at 3:30
@rm-rf You may well be right - but I thought it's OK to postulate "innocent until proven guilty"... – Jens Apr 1 '14 at 3:35
In other words, D[f[x]] is the zeroth derivative, $f^(0)(x)$, the same as Derivative[0][f][x]. – Michael E2 Apr 1 '14 at 15:48
@MichaelE2 Right. I wonder if there's also a continuation to negative derivatives like in Derivative[-1], which is indeed the integral. Maybe f[x[D]]? – Jens Apr 1 '14 at 16:36
@Jens I'm not sure about f[x[D]]; however, I'd expect D[f[x], {x,-1}] would be ok, but it doesn't work. – Michael E2 Apr 1 '14 at 16:51