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Basically, I noticed that if you multiply by x or by y in an equation before implicitly deriving, you get two different answers. Unfortunately their whole discussion about level sets, I'd have to sit down for hours to try and understand since my multivariable calc is not the best, neither is my normal calc but when they started talking about level sets I realized I'd have to sit there really long to understand that thoroughly. Maybe I will if I can't get the response I'm looking for here, but I'm thinking maybe someone can enlighten me quicker.

So I was wondering if anyone can help me with this, why do the derivatives produce different results, specifically, for example, for that problem in the thread? Even further, what about the popular logarithmic differentiation? If you had for example x^3y+y^2=8y, and you do logarithmic differentiation vs. just differentiating it, there is also a different answer for y', and plugging in x,y coordinates give different results for the different y'.

Thanks for any replies!