Help needed with partial differentiation equation of matrices

[Bignose]

I would appreciate it if someone could provide some helpful inputs to my original question. The equations in my question can be found in this reference Journal of Applied Physics 79, 7148 (1996). Thanks in advance.

mikeraj, it is obvious that there is a typo somewhere. Either in your transcription, or in the original paper. Because the differentiation in the equation of constant vectors and with respect to constant vectors is meaningless. It is not defined. It doesn't exist. As zztop points out, it does appear that if you ignore the differentiation symbols, and perform some kind of tensor/element-by-element division, then you get the intended answer. It is certainly possible that the symbols were accidentally left in or added to the manuscript. Another possible avenue to seek help may be to seek out if an errata has been published to that paper, or possibly even reach out to the authors.