So if I have the curve $ y = x $ and I do implicit differentiation, I get $ y' = 1$
But when I do implicit differentiation of $ \frac{y}{x} = 1$ I get $y' = \frac{y}{x}$
I guess on all points on $ y = x$ we get the same derivative but I still don't fully understand how to understand getting two different derivatives for the same curve (almost the same curve except for $ x \neq 0$)
More generally, for any given curve of the form $\frac{f(x,y)}{g(x,y)} = h(x,y)$ I am getting different answers when I do implicit differentiation of $ f(x,y) = h(x,y)g(x,y)$ vs the original form.
For instance, implicit differentiating $\frac{2x-y^3}{y+x^2} = 5x-4$ gives me $y' = \frac{-5x^4 - 2x^2 (5y+1)+2xy^3 + (2-5y)y}{3x^2y^2 + 2x+2y^3}$
Whereas implicit differentiating $2x - y^3 = (5x-4)(y+x^2)$ gives me $y' = \frac{-15x^2 + 8x -5y +2}{5x+3y^2-4}$
Which is the right way to go about it and more importantly what is going on ? How am I supposed to understand the different answers I get from implicit differentiation before vs after I perform some cross multiplications?