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I have a question on an implicit differentiation problem. I get two different answers depending on how I do it and the answers are different (not just looking, but different).

The problem is [tex] \frac{x+3}{y}-4x-y^2=0 [\tex]. One option is to just differentiate as it stands and you get [tex] y'=\frac{y-4y^2}{x+3+2y^3} [\tex]. Another possibility is to initially multiply both sides of the equation by y to simplify it. If you do that the derivative you obtain is [tex] y'=\frac{1-4x}{4x-3y^2} [\tex]. These are two different derivatives but I'm not sure why this happens.

Any thoughts would be appreciated (NOTE: I'm teaching the course and this came up).

P.S. why aren't my equations tex-ing?

Kevin