In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus.
The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.
If G is a free group with identity element e and generators gi, then the Fox derivative with respect to gi is a function from G into the integral group ring
, and obeys the following axioms: The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule.
As a consequence of the axioms, we have the following formula for inverses The Fox derivative has applications in group cohomology, knot theory, and covering space theory, among other areas of mathematics.