How to deal with the total differential of implicit function equation

Question

I want to find the total differential of $z=z(x, y)$, $z=z(x, y)$ satisfies the implicit function equation $(x+1) z-y^{2}=x^{2} f(x-z, y)$ (function $f(u, v)$ is differentiable).
Dt[(x + 1) z[x, y] - y^2 == x^2*f[x - z[x, y], y], z[x, y]]

But the above result is not in the form of $mathrm{d} z=p(mathrm{x}, mathrm{y},mathrm{z(x,y)}) mathrm{d} mathrm{x}+mathrm{q}(mathrm{x}, mathrm{y},mathrm{z(x,y)}) mathrm{d} mathrm{y}$.
What should I do to get the form I want?
Test examples:
$left.boldsymbol{d} zright|_{(0,1)}=-boldsymbol{d} x+2 boldsymbol{d} y$

march · Accepted Answer

I would do it the following way.  Don't define the function z as a function of x and y:
expr = (x + 1) z - y^2 == x^2*f[x - z, y];
exprD = Dt@expr

results in

Then, solve for Dt[z], collect in the variables Dt[x] and Dt[y], and use the third argument to Collect to replace the x and y without replacing the values in the Dt[x] and Dt[y]:
Collect[
  Dt[z] /. First@Solve[exprD, Dt[z]],
  {_[x], _[y]},
  # /. {x -> 0, y -> 1} &
 ]
(* -z Dt[x] + 2 Dt[y] *)

How to deal with the total differential of implicit function equation

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