> But `x^(2y) + y^(x^2)` is not a function; neither syntactically (since it refers to both x and y) nor functionally (since the relation ignores the sign of x).

> But `x^(2y) + y^(x^2)` is not a function; neither syntactically (since it refers to both x and y) nor functionally (since the relation ignores the sign of x). It is a function, if you interpret it as f(x) = x^(2y(x)) + y(x)^(x^2). Of course, we don't know what kind of function y is. And if y is given implicitly, there might be multiple possible ys. But that doesn't stop us from manipulating the composite function. Specifically, if there exist multiple ys, then the manipulated equations will be valid for all of these ys. So the only extra step is thinking about functions abstractly, instead of being given concrete functions. But otherwise it really is the "normal rules of calculus.