The Fox derivative is the 
-linear function 
from the integral group
ring of a free group to itself specified on the
generators by 
and on products of group elements 
and 
by 
.
Here 
is the Kronecker delta, and the derivative is
extended to the group ring by 
-linearity. Fox derivatives turn group
presentations into matrices used to compute Alexander
invariants.
Fox Derivative
See also
Alexander Polynomial, Free Group, Group RingExplore with Wolfram|Alpha
References
Fox, R. H. "Free Differential Calculus. I. Derivation in the Free Group Ring." Ann. Math. 57, 547-560, 1953.Cite this as:
Weisstein, Eric W. "Fox Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FoxDerivative.html