Notation for dy/dx in multivariable calculus. I am fairly new to calculus (I a learned a bit from the interwebs, not studied it), but I wanted to move to multivariable calculus because while making a 3D rendering program, order of rendering was not quite right, so my guess was to render in order based on the distance. It seems to work, but not as perfectly as I would want, so I tried to use my calculus knowledge to find the closest distance between a triangle and origin or $(0,0,0)$.
My question is, because I dont want to waste 2 hours doing math wrong, when in regular calculus you would note it
$$
    \frac{dy}{dx} = \lim_{h \to 0} \left(\frac{f(x+h)-f(x)}h\right)
$$
then how would you note that properly with say 2 variables? Or am I not supposed do it like this but differently?
I was thinking about
$$
    \frac{dy}{dx+dz} = \lim_{(h_x,h_z)\to(0,0)}\left(\frac{f(x+h_x,z+h_z)-f(x,z)}{h_x+h_z}\right)
$$
but I dont know if that makes sense.
I know that you can write $x$ and $z$ as a function of $t$ like
$$
    \frac{dy}{dt} = \lim_{h\to0}\left(\frac{f(x(t+h), z(t+h)) - f(x(t), z(t))}{h}\right)
$$
but this makes it regular calculus and i am not able to make a function that would return all points on a triangle with just a single variable. (I do that by slicing the triangle to infinitely many lines, pick one using $x$ and slice that line into infinitely many points and pick one using $z$. Final goal is to find the minima in interval from $[0,0]$ to $[1,1]$.)
Just to clarify: my question is just about noting this properly, not about the answer to the problem I'm tring to solve with it.
 A: Typically, the in three variable function, you denote the partial derivatives as
 $\frac{\partial f}{\partial x}= \lim_{h\to 0}\frac{f(x+ h, y, z)- f(x, y, z)}{h}$
 $\frac{\partial f}{\partial y}= \lim_{h\to 0}\frac{f(x, y+ h, z)- f(x, y, z)}{h}$
 $\frac{\partial f}{\partial z}= \lim_{h\to 0}\frac{f(x, y, z+ h)- f(x, y, z)}{h}$
In general $R^n$ and $R^m$ we can take "the derivative" of a function, f, from $R^n$ to be $R^m$  to be "the linear transformation from $R^n$ to $R^m$ that best approximates f at the given point.  In particular, a differentiable function, f(x,y,z), from $R^3$ to R, can be approximated, in a neighborhood of $(x_0, y_0, z_0)$ as $f(x,y,z)= f(x_0,y_0,z_0)+ L(x_0, y_0, z_0)\cdot (x- x_0, y-y_0, z-z_0)+ \Delta(x,y,z)$ where $L(x,y,z)$ is that linear transformation and $\Delta(x,y,z)$ is a function  such that $\lim_{(x,y,z)\to (x_0,y_0,z_0)} \frac{\Delta(x,y,z)}{\sqrt{(x-x_0)^2+ (y-y_0)^2+ (z-z_0)^2}}= 0$.  It follows that that "linear transformation", $L(x,y,z)$ is the gradient $\nabla f= \left<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial x}\right>$.  
In that sense, we can think of the gradient at each point as being the "derivative" there
