If v runs parallel to one of the axes the derivative is called a partial derivative. In R2 there are two axes, traditionally called the x and y axes, hence there are two partial derivatives. They represent the change in f as you walk due east, or due north, parallel to the positive x and y axes respectively. If you are walking east, the directional derivative is called the partial of f with respect to x, and is written f∂x. The partial with respect to y is written f∂y. Think of the operator ∂ as shorthand for "differentiate with respect to".
Let's compute some partials, using the definition of directional derivative. When computing f∂x, the direction vector is 1,0, hence all of h is added to x, and none of h is added to y. Since the y coordinate doesn't change, you can treat it as a constant, like 3 or 17. You only need differentiate with respect to x, as though it were a function of x. Similarly, x is treated as a constant when computing f∂y. The partials for 4xy+3x+2y+1 are 4y+3 and 4x+2, respectively.
Realize that one can derive a formula for the partial derivative across the entire domain in one go, as we did in the above example. This transforms the original function into a new function that gives the slope, in the x or y direction, at every point.
We can then take partial derivatives of this new function. The result is called a second partial, and is similar to a second derivative. If both partials are taken with respect to x, the result is denoted f∂x∂x, and indicates the curvature of the surface in the x direction. Like the second derivative on a graph, the second partial tells us how the slope, in the x direction, is changing as we move east. Similarly, f∂y∂y tells us how the slope, in the y direction, is changing as we head north. The mixed partial f∂x∂y tells us how the east-west slope changes as we walk north. Place a level on the surface, oriented east-west, and watch how it tilts up and down as you slide it to the north. We will see that f∂x∂y = f∂y∂x, but it will take some effort to prove that.
Partials can be generalized to any number of variables, as in f∂x∂y∂z in three dimensions.
It would be nice if the existence of directional derivatives along all unit vectors implied continuity; it does not. Consider f(x,y) = xy2 over x2+y4, with f(0,0) = 0. Follow the parabola x = y2 as it curves into the origin, and f equals 1/2. Yet f is 0 at the origin. Replace y with mx to look at the directional derivative along the line with slope m. This gives m2x over 1+m4x2, which is differentiable at x = 0. (Technically m should be scaled to represent a vector of unit length, but we're only trying to show differentiability here; we don't need the precise value.) One direction that has not been analyzed is the partial with respect to y, where m would equal infinity. Since f is 0 along the y axis, this partial is 0. Every directional derivative is well defined, but as we saw earlier, f is not continuous.
The problem is that the surface is not smooth, even at the microscopic scale. An insect the size of a proton would still see a hill rising up in the first quadrant, even though the surface is flat along the axes. We don't consider a surface, even a continuous surface (which this one is not) to be differentiable if it wiggles up and down at the smallest scales. We want it to look smooth, like a (possibly tilted) plane.
The function f(x,y) = xy is zero down the axes, with a hill up the first quadrant, but at the smallest scale the surface looks flat at the origin. We can approximate it by a flat plane, and the approximation is pretty good, at least for a while.