  Derivations provide a way of transporting ideas from the calculus of
manifolds to algebraic settings where there is no sensible notion of limit. In
this paper, we consider derivations in certain monoidal categories, called
codifferential categories. Differential categories were introduced as the
categorical framework for modelling differential linear logic. The deriving
transform of a differential category, which models the differentiation
inference rule, is a derivation in the dual category. We here explore that
derivation's universality.
  One of the key structures associated to a codifferential category is an
algebra modality. This is a monad $T$ such that each object of the form $TC$ is
canonically an associative, commutative algebra. Consequently, every
$T$-algebra has a canonical commutative algebra structure, and we show that
universal derivations for these algebras can be constructed quite generally.
  It is a standard result that there is a bijection between derivations from an
associative algebra $A$ to an $A$-module $M$ and algebra homomorphisms over $A$
from $A$ to $A\oplus M$, with $A\oplus M$ being considered as an infinitesimal
extension of $A$. We lift this correspondence to our setting by showing that in
a codifferential category there is a canonical $T$-algebra structure on
$A\oplus M$. We call $T$-algebra morphisms from $TA$ to this $T$-algebra
structure Beck $T$-derivations. This yields a novel, generalized notion of
derivation.
  The remainder of the paper is devoted to exploring consequences of that
definition. Along the way, we prove that the symmetric algebra construction in
any suitable symmetric monoidal category provides an example of codifferential
structure, and using this, we give an alternative definition for differential
and codifferential categories.
