Suppose I wish to compute the implicit derivative of $\sqrt{x^2+y^2}=x+y$. One could differentiate both sides with respect to $x$, yielding $y\prime$ which we can make the subject.
Say I were to first square both sides, getting the expression $x^2+y^2=(x+y)^2$, and then I were to proceed as usual.
My question is, is this a valid method of computing the implicit derivative? Can squaring both sides cause problems?
In general, it can cause minor problems, since the squared equation isn't the same as the original. The simplest example:
$$x = y$$ isn't the same as
$$x^2 = y^2$$ since the latter has solutions $$x = \pm y$$
In the equation in your question there isn't a problem because the LHS is never negative, assuming that you're following the usual convention that $\sqrt{x}$ denotes the non-negative square root of $x$.
However, that equation is a degenerate hyperbola, so its derivative is a bit funny anyway.