Do things get more complicated in higher dimensions? Not really.
The important thing to realize about a partial derivative, is that it is by its definition, an ordinary derivative, albeit one in which certain dependences are ignored, and it is computed exactly as ordinary derivatives are computed. There are no new tricks and none are needed.
There are some issues that arise when some quantity depends on some variables, which in turn depend on others.
Suppose, for example that we are interested in the temperature T of a tiny
body that is moving through ordinary space with the property that as it moves,
its temperature assumes that of its surroundings. This temperature varies in
time and also in space.
The temperature of the body will then change because of the change in T with time, but also because of its motion.
Here T in space is a function of position (x, y, z) and time t: T = T(x, y, z, t) (we use the same letter to describe both time and temperature to maximize confusion in what otherwise would be a bald and unconvincing narrative).
Now suppose further that the body in question has a trajectory through space
described by equations x = x(t), y = y(t), z = z(t). (You might want to abbreviate
this as r = r(t) with r = (x, y, z).)
We raise the question, what is the derivative with respect to time of the temperature experienced by that body?
We write dT out in terms of differentials
We also have
Putting these together we get
from which we conclude
It is not a bad idea to realize whenever you encounter a formula that looks like this that it undoubtedly arises from an analogous situation, when a function depends on time and also upon spatial variables that themselves depend on time.
This kind of thing is sort of a generalized chain rule and is sometimes referred to as such.
Please notice that the way to handle any and all problems of this kind is to consider differentials, include in their relations to one another all possible dependencies, and relate them all to the differential of the independent variable, here t. You can then divide the differentials to find the derivative.
There is a complication that arises when there are several variables that are interrelated. Then there can be different partial derivatives, depending on which other variables are kept constant and you may have a choice as to fix which coordinate when you very a particular one of them. To keep things straight you must introduce a notation that has a place where you can describe which variable(s) are to be kept fixed.