微分可能な関数fに対して、∀xでf(x) = g(x), f'(x) = g'(x)となるgを構成できますか？

x = X - Yf'(x_0) + x_0
y = Xf'(x_0) + Y + f(x_0)

d(y - f(x))|(x, y) = (x_0, f(x_0))
= dy - df(X - Yf'(x_0) + x_0)
= f'(x_0)dX + dY - f'(x_0)dX + f'(x_0)^2dY
= (1 + f'(x_0)^2)dY