Ordinary/Partial Differentiation Confusion So, I've seen from different sources that $\frac{d}{dx}$ and $\frac{\partial}{\partial x}$ function as basically the same thing. But then, I saw that if $$\frac{d}{dx}(z(x, y(x)) = 0$$ then $$ z(x, y(x)) = c$$
where c is a constant, because "if the ordinary (not partial…) derivative of something is zero, that something must have been a constant to start with."
Related to this I think is the notion that in implicit differentiation, taking the $\frac{d}{dx}$ of a given $g(y)$ gets $\frac{d}{dy}\frac{dy}{dx}$, while in other cases I've heard that taking $\frac{d}{dx}$ of $g(y)$ simply equals zero since y can be treated as a constant.
What are the distinctions here?
 A: $\frac{d}{dx} z(x, y(x))$ is unambiguous: we have a function $f(x) = z(x, y(x))$ of $x$ obtained by composing two other functions $z$ and $y$, and we want to compute its derivative. But you might interpret the partial derivative
$$\frac{\partial}{\partial x} z(x, y)$$
to refer to the partial derivative of $z$ in the first variable only, holding $y$ fixed, and ignoring the fact that $y$ is a function of $x$. You might do this in order to state the chain rule in this setting, for example, which can be written
$$\frac{d}{dx} z(x, y(x)) = \frac{\partial}{\partial x} z(x, y) + \frac{\partial}{\partial y} z(x, y) \frac{dy}{dx}.$$
This gets at a very ambiguous thing about partial derivative notation, which is that any partial derivative is implicitly measuring the change in one variable while "keeping all other variables fixed," and it can get quite ambiguous what "all other variables fixed" means; in particular, although the notation only refers to a single variable it actually depends on a choice of many other variables (the "fixed" ones), which is omitted from the notation and to be "understood from context."
This can get very confusing and in my opinion is a sign that partial derivative notation should be replaced by something more precise, although it's tricky to see how to do it without making everything excessively verbose. I think Sussman and Wisdom's Structure and Interpretation of Classical Mechanics may do something like this but I haven't read it.  They at least discuss the example of the Euler-Lagrange equation in the preface.
