  In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld
additive') analogs each, for the well-known basic congruences of Fermat and
Wilson, in the case of polynomials over finite fields. When we look at them
modulo higher powers of primes, i.e. at `supercongruences', we find interesting
relations linking them together, as well as linking them with arithmetic
derivatives and zeta values. In the current work, we expand on the first analog
and connections with arithmetic derivatives more systematically, giving many
more equivalent conditions linking the two, now using `mixed derivatives' also.
We also observe and prove remarkable prime factorizations involving derivative
conditions for some fundamental quantities of the function field arithmetic.
