An n-ellipse is a generalization of a circle with $n$ foci. It can be written as $\sum\sqrt{(x-a)^2 + (y-b)^2} = R$. I am curious as to the derivative $\frac{dy}{dx}$ of this relation.
Using implicit differentiation and differentiating the left side term by term, I found the solution to be $\sum[(x-a)+(y-b)\frac{dy}{dx}] = 0$, or $ \frac{dy}{dx} = -\frac{\sum(x-a)}{\sum(y-b)}$. This is a very simple variation on $\frac{dy}{dx}$ of a circle.
However, the behavior of $\frac{dy}{dx}$ is different in an n-ellipse than a circle, leading me to believe my differentiation is incorrect. In a circle, $\frac{dy}{dx}$ is independent of radius. However, in a 3-ellipse, changing the "radius" on the RHS changes the derivative. What's wrong with my derivation or interpretation?
I think you are assuming that
$\sum_{i=1}^n\sqrt{(x-a_i)^2 + (y-b_i)^2} = R \space \Rightarrow \space \sum_{i=1}^n(x-a_i)^2 + (y-b_i)^2 = R^2$
which is incorrect (unless $n=1$).