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.

Average Slope vs First Derivative

Average Slope versus the first derivative

The website below provides a nice, clear and straightforward discussion on the difference between average slope and the first derivative. I have attempted to summarize some of the key points.

https://www0.gsb.columbia.edu/premba/analytical/s4/s4_4.cfm


Finding slope of a curve

1. Draw a line through two points on the curve which are very close to the selected point  and then measure the slope of the line drawn. The closer the two points are together, the closer the the slope of the secant line is to the actual slope of the curve at the point of interest.

2.  Draw a line tangential to the point of interest on the curve.  The tangent line's slope is called the "instantaneous rate of change" because it is a precise measurement of the slope of the curve at that particular point. Use the first derivative of the nonlinear function to find the slope of the tangent line at any point on the curve without having to find the equation of the tangent line.

The derivatives of some commonly encountered quantities are:



http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/BASICMAT/node5.html



Finding Minimum and Maximum
The maximum and minimum points can be found by taking the first derivative of the function, setting it equal to zero, and solving for x.