  One of the fundamental tools of undergraduate calculus is the chain rule. The
notion of higher order directional derivatives was developed by Huang,
Marcantognini, and Young, along with a corresponding higher order chain rule.
When Johnson and McCarthy established abelian functor calculus, they proved a
chain rule for functors that is analogous to the directional derivative chain
rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an
analogue of the iterated directional derivative and provided an inductive proof
of the analogue to the chain rule of Huang et al.
  This paper consists of the initial investigation of the chain rule found in
Bauer et al., which involves a concrete computation of the case when $n=2$. We
describe how to obtain the second higher order directional derivative chain
rule for abelian functors. This proof is fundamentally different in spirit from
the proof given in Bauer et al. as it relies only on properties of cross
effects and the linearization of functors.
