First and Second Derivatives on Rasters

Four topographic indices that require the use of derivatives are:
- slope
- aspect
- profile curvature
- plan curvature

Although it is taught that rasters model continuous surfaces, in practice, rasters are made up of many discrete tiles. So, to get the two points needed to measure slope, we have to rely on points located in neighboring cells.  Typically, use is made of 4 or nine neighboring cells. Slope is calculated as the sum of the change in elevation with respect to distance along both the x-axis and the y-axis, as shown in the formula below.

Aspect is calculated as:

Profile and plan curvature are calculated as second derivatives of the slope.

The question I am puzzled about is whether a true derivative can be taken on this gridded raster surface. The two formula above applies to average slope, so that is clear, however, I can't see  how instantaneous rate of change applies.