My question is simple.
We are given the following function: $\frac{x^2-y^2}{x^2+y^2}=\frac{1}{2} $.
We are asked to find the derivative implicitly.
If we use the chain rule on the left-hand side, we can solve for $\frac{dy}{dx} = \frac{y}{x}$.
But if we multiply out the denominator first and then take the derivative, we end up with a different result, namely $\frac{dy}{dx}=\frac{x}{3y}$.
What is the reason? What is the theory behind this duplicity?