Within this brief text starting on page 2, the author(s) have a habit of employing a sort of "mixed" notation of differentiation, mixing the "prime marks" of Lagrange's notation for differentiation of a single-variable with the subscript notation commonly used in partial differentiation; that is, something like:
f 'x (x0,y0)
Now, being that this text has had a few errors that I have noticed already, this could simply be an accidental use of the Lagrange notation via force of habit for performing differentiation; however, being that this "error" is repeated throughout the document, I find this unlikely to be the case. Might it be that this is some sort of abuse of notation used to represent a more common mathematical concept? If so, what mathematical concept might that be?