Implicit differentiation question Given: yx−y=x2+1

${\displaystyle \frac{y}{x-y}=x^2+1}$
You claim that ${\displaystyle y^{\prime }=2x}$
so that ${\displaystyle y=x^2+C}$
This means ${\displaystyle \frac{x^2+C}{x-x^2-C}=x^2+1}$
This is absqrt, since the quotient of two second degree polynomials can't be a second degree polynomial. In fact you get two non vanishing terms ${\displaystyle x^3\text{and}{x}^4}$ which are off.
I don't understand what your procedure is, also. I would proceed as follows:
${\displaystyle \frac{y}{x-y}=x^2+1}$
${\displaystyle \frac{d}{dx}\left(\frac{y}{x-y}\right)=\frac{d}{dx}(x^2+1)}$
${\displaystyle \frac{y^{\prime }(x-y)-(1-y^{\prime })y}{{(x-y)}^2}=2x}$
${\displaystyle \frac{y^{\prime }x-y}{{(x-y)}^2}=2x}$
${\displaystyle y^{\prime }x=2x{(x-y)}^2+y}$
${\displaystyle y^{\prime }=2{(x-y)}^2+\frac{y}{x}}$

An explicit approach:
Rewrite as ${\displaystyle y=(x-y)(x^2+1)}$, and factor out y to get ${\displaystyle y=\frac{x^3+x}{x^2+2}}$, This is straightforward to differentiate, yielding ${\displaystyle \frac{dy}{dx}=\frac{x^4+5x^2+2}{{(x^2+2)}^2}}$.