I'm sorry if I sound too ignorant, I don't have a high level of knowledge in math.
I'm just starting Calc II, yet, I have a tiny understanding of derivatives in higher dimensions.
Having $z(x,y)$, one could treat $y$, for instance, as a constant, in order to get a function with respect to $x$ and then differentiate as one would do in two dimensions. This would, intuitively, give the derivative in the direction of $x$. To my understanding, differentiating with respect to $x$ is different as differentiating with respect to $y$, unless the function $z$ has some sort of symmetry.
But recently I have read in math posts that functions of more than one variable have only one derivative at each point, and that what I have described above is merely the 'directional derivative'.
If this is true, what is this type of derivative they are talking about? What is the equation for it?
Any help/thoughts would be really appreciated.