How can I numerically evaluate the total derivative of a multivariate function? I think I understand now the intuitive reasoning behind the total derivative of a multivariate function $z = z(x, y)$, which is 
$$
dz = \frac{\partial{z}}{\partial{x}}dx + \frac{\partial{z}}{\partial{y}}dy
$$
So let's take an example, $z = x^2 + y^2$, a paraboloid and a surface of revolution. Here $z_x = 2x$ and $z_y = 2y$, so
$$
dz = 2xdx + 2ydy.
$$
How would I evaluate this numerically? For example, if I had a program that calculated the infinitesimal change $dz$ at a point $\langle x_0, y_0 \rangle$, I could plug in the values of $x_0$ and $y_0$ for $x$ and $y$ respectively, but what would I plug in for $dx$ and $dy$?
EDIT: My guess is that you actually can't evaluate it directly as a number, but instead have to use it as a combination of infinitesimals, kind of like how you can't treat problems involving the imaginary unit $i$ as all real numbers. So instead of treating $dz$ as some actual value, you use it to derive other values that are useful. That's my guess but I'm not sure if it's valid or not.

The differential is not a numerical value, so it's meaningless to associative a number to a differential. But it indeed has a representation using a set of basis, in which the differential is regarded as some kind of vector. Say instead we use binary tuple $(2x,2y)$ to represent differential $dz$. In practice, we also don't care about $dz$. What we concern is how it applies to a vector $w=(u,v)$. So now we can give $dz(w)$ a numerical value by $2(ux+vy)$.
