We compute two-sided second-order epi-derivatives for certain composite fun
ctionals f = g o F where F is a C-1 mapping between two Banach spaces X and
Y, and g is a convex extended real-valued function on Y. These functionals
include most essential objectives associated with smooth constrained minim
ization problems on Banach spaces. Our proof relies on our development of a
formula for the second-order upper epi-derivative that mirrors a formula f
or a second-order lower epi-derivative from [7], and the two-sided results
we obtain promise to support a more precise sensitivity analysis of paramet
erized optimization problems than has been previously possible.