the problem is $ x^2=(x+y)/(x-y)$
If I start the problem by doing implicit differentiation on both sides I get a different answer than if I first cross multiply and start with $ x^3-x^2y=x+y $.
Is there any mathematical reason that the two methods would not yield the same result? With cross multiplication there is a $3x^2$ term which does not appear in the final answer if you start with implicit differentiation of both sides.
Best Answer
They are not different.
Let $$z(x,y)= x^2 - (x+y)/(x−y)=0 \tag{1}$$
Then we get $$\frac{dy}{dx}=-\frac{\frac{\partial z}{ \partial x}}{\frac{\partial z}{ \partial y}}= \frac{x\,{{y}^{2}}+\left( 1-2{{x}^{2}}\right) y+{{x}^{3}}}{x} \tag{2} $$
Let $$w(x,y)= x^3 − x^2y - x-y=0 \tag{3}$$
$$\frac{dy}{dx}=-\frac{\frac{\partial w}{ \partial x}}{\frac{\partial w}{ \partial y}}=-\frac{2xy-3{{x}^{2}}+1}{{{x}^{2}}+1} \tag{4}$$
Now, expressions $(2)$ and $(4)$ might seem different, but you must remember that $x,y$ are not free, the restrictions $(1)$ or $(3)$ apply. In this case we can isolate $y=(x^3-x)/(x^2+1)$, and replacing we can check that both $(2)$ and $(4)$ are equal:
$$\frac{dy}{dx}=\frac{{{x}^{4}}+4{{x}^{2}}-1}{{{x}^{4}}+2{{x}^{2}}+1}$$