Calculate vertical gradient of velocity

[zzyfathead]

Hello,
I am trying to calculate vertical gradient du/dz and dv/dz from ROMS output. Usually, we use differential equation shown in picture below. However, it may lead some non-conservative problems, and I dont know how to deal with the the upper half part of the layer 8 in the picture (Red circle). Could ROMS output gradient terms by itself?
Thanks for answering this question which may be easy for you.

zzy

[rduran]

the endpoints are not differentiable, the definition of derivative at a point requires a continuous function in the vicinity of the point, in this case, the function to be differentiated is not even defined in the vicinity of the point, let alone continuous and with a well-defined slope. The best you can do is compute a one-sided derivative but in this case that is effectively the same as extrapolating the derivative you can compute and that you illustrate—it is not the same as the derivative in the formal sense.