  Differential categories were introduced by Blute, Cockett, and Seely as
categorical models of differential linear logic and have since lead to abstract
formulations of many notions involving differentiation such as the directional
derivative, differential forms, smooth manifolds, De Rham cohomology, etc. In
this paper we study the generalization of differential algebras to the context
of differential categories by introducing $\mathsf{T}$-differential algebras,
which can be seen as special cases of Blute, Lucyshyn-Wright, and O'Neill's
notion of $\mathsf{T}$-derivations. As such, $\mathsf{T}$-differential algebras
are axiomatized by the chain rule and as a consequence we obtain both the
higher-order Leibniz rule and the Fa\`a di Bruno formula for the higher-order
chain rule. We also construct both free and cofree $\mathsf{T}$-differential
algebras for suitable codifferential categories and discuss power series of
$\mathsf{T}$-algebras.
