How to write a partial derivative for when two variables change simultaneously? Suppose that I have a function $y(x,z)$ and any change in $x$ coexists with a change in $z$ (e.g., $x$ goes "up", $z$ goes "down").
What would be the partial derivative notation for such a case? Would it still just be with respect to one variable?
 A: You seem to say $x$ might be a function of $z$.  Say $x(z)$.
Then you could do something like $y(x,z)=y(x(z),z)$, so that you have a single variable function of $z$.
Then you could take the $z$ derivative:
$\frac{\mathbb dy(x,z) }{\mathbb d z}$.
But without knowing what $y$ is, I won't take it any further.
When you learn the $\bf{chain\ rule}$, you will learn how to deal with this situation.
A: It is very hard for me to give you a clear answer without giving an example. But given what we got, it sounds like both x and z are dependent/parametrized by 1 other variable, lets call it $t$.
My guess is that what you want is the partial derivative of y with respect to $t$.
This can be done using the chain rule.
Edit:
If we have:
$$
y = f(x,z)
$$
$$
x = f_{1}(t)
$$
$$
z = f_{2}(t)
$$
Then what you want to do is to just substitute $x$ and $z$ in terms of $t$ into $f(x,z)$ and find $\frac{dy }{dt}$, it wouldn't be a partial derivative anymore. However, you can still find:
$$
\frac{\partial y}{\partial t \partial x} \ AND \  \frac{\partial y}{\partial t \partial z}
$$
In many situations those 2 are the same.
