  Starting with a non-abelian gerbe represented by a non-abelian differential
cocycle, with values in a given crossed-module, this paper explicitly
calculates a formula for the derivative of the associated surface holonomy of
squares mapped into the base manifold; with spheres later considered as a
special case. While the definitions in this paper used for gerbes, their
connections, and the induced holonomy will initially be simplicial,
translations into a cubical setting will be provided to aide in explicit
coordinate-based calculations. While there are many previously published
results on the properties of these non-abelian gerbes, including some
calculations of the derivative over a single open set, this paper endeavors to
take these local calculations and glue them together across multiple open sets
in order to obtain a single expression for the change in surface holonomy with
respect to a one-parameter family of squares.
