Notation for implicit derivative

Question

Consider the following line of code:

D[x == y^3 + x y, x, NonConstants -> y]

The output would be:

1 == y + x D[y, x, NonConstants -> {y}] + 3 y^2 D[y, x, NonConstants -> {y}]

This is a confusing and cumbersome notation for the more natural:

1 == y + x y' + 3y^2 y'

I am trying to use the Notation package to help me replace the messy, default, output with the more natural one.
I have not succeeded.

Of-course I have read this Q. The answer, if relevant, seems inaccessible to me, unfortunately.
Any help would be appreciated.

Answer 1

Based on the comments above I believe you are merely looking for a way to automate Artes' replacement. Using $PrePrint:

$PrePrint = # /. {D[y_, x_, NonConstants -> {y_}] :> y'[x]} &;

Now:

D[x == y^3 + x y, x, NonConstants -> y]

1 == y + x y'[x] + 3 y^2 y'[x]

You could also use $Post or Format or MakeBoxes depending on your specific needs.
Using MakeBoxes is usually robust and does not tie up $Post or $PrePrint.

MakeBoxes[D[y_, x_, NonConstants -> {y_}], fmt_] := ToBoxes[y'[x], fmt]

Answer 2

I would consider using Dt:

Dt[x == y^3 + x y, x]
(* 1 == y + x Dt[y, x] + 3 y^2 Dt[y, x] *)

You can specify literal constants with SetAttributes:

ClearAll[a];
SetAttributes[a, Constant];
Dt[(x^2 + y^2)^2 == a^2 x y, x]
(* 2 (x^2 + y^2) (2 x + 2 y Dt[y, x]) == a^2 y + a^2 x Dt[y, x] *)

If x and y might become dependent variables, then omit the second argument and you get:

de = Dt[(x^2 + y^2)^2 == a^2 x y]
(* 2 (x^2 + y^2) (2 x Dt[x] + 2 y Dt[y]) == a^2 y Dt[x] + a^2 x Dt[y] *)

You can turn y into an explicit function of x:

de /. y -> y[x]
(* 2 (x^2 + y[x]^2) (2 x Dt[x] + 2 Dt[x] y[x] y'[x]) == a^2 Dt[x] y[x] + a^2 x Dt[x] y'[x] *)

If you want to turn x into the dependent variable, you can replace Dt[x] by 1:

de /. y -> y[x] /. Dt[x] -> 1
(* 2 (x^2 + y[x]^2) (2 x + 2 y[x] y'[x]) == a^2 y[x] + a^2 x y'[x] *)