To differentiate an implicit function $y(x)$, defined by an equation $R(x, y) = 0$ one can totally differentiate $R(x, y) = 0$ with respect to $x$ and $y$ and then solve the resulting linear equation for $\frac{dy}{dx}$ to explicitly get the derivative in terms of $x$ and $y$.
Consider the following example: Let $y(x)$ be defined by the following relation:
$$(x^2-y^2)^{1/2}+\arccos\frac{x}{y}=0. \,(y\neq 0.)$$
Clearly, the equation defines $y$ as a function of $x$. In fact, it is easy to see that $y=x$. However, when I apply the method of implicit differentiation to $(x^2-y^2)^{1/2}+\arccos\frac{x}{y}=0$, I failed to get the desired result $\frac{dy}{dx}=1$ (since $y=x$). Why does implicit differentiation fail here?
Edit: I did not do the implicit differentiation by hand as it is too tedious; instead I trusted the result on WolframAlpha: