Convert the contours to a surface using the Topo to Raster tool found in the Interpolation Toolset in either 3D Analyst or Spatial Analyst toolbox. In other words average gradient is the same as the rate of change. This is shown by the two-point equation for a straight line Section 12.
First create a slope raster from your DEM using Slope Spatial Analyst tool.
At point A h 0 and a 1. Once you have the slope raster you can use Zonal Statistics as Table Spatial Analyst tool to calculate MEAN values of slope for each Zone parcels or storm water route. Luckily we dont have to calculate the slope manually. Determine the corresponding y-coordinates.