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The code

x = var('x')
y = function("y")(x)
eq = x*e**y == x - y
dy = diff(y,x)
sol = solve(diff(eq), dy)
show(dy.subs(sol))

yields $$-\frac{e^{y\left(x\right)} - 1}{x e^{y\left(x\right)} + 1}$$ as expected