  We explore the notion of a category with coproducts $\cup$ and a monoidal
structure $\otimes$ distributing over it, endowed with an endo-functor
$\partial$ which is "linear and Leibniz". Such $\partial$ can be legitimately
called a *derivation* on $\mathcal{C}$, and the pair $(\mathcal{C},\partial)$
appears as the categorification of a differential ri(n)g: a differential 2-rig.
  We explore a number of possible special examples: when $\mathcal{C}$ is
semicartesian (i.e., its $\otimes$-monoidal unit is the terminal object),
derivations are better behaved in some respect; interesting examples arise when
$\mathcal{C}=[\mathcal{A},\textsf{Set}]$ is a presheaf topos endowed with a
convolution product: among these, the operation of Brzozowski's derivative in
formal language theory; we retrieve the "chain rule" in a category of
"non-symmetric species" $[\textsf{Fin},\textsf{Set}]$, obtained when a
substitution product $F\diamond G$ is interpreted as composition $T_F\circ T_G$
of the associated finitary functors, regarded as formal power series.
  Derivations on a 2-rig are tightly related to *tensorial strengths*; this
draws a connection between the present theory and applicative structures in
functional programming. Moreover, given a derivation on $\mathcal{C}$, the
derivative of the $\otimes$-monoidal unit $\partial I$ is a *self-similar*
object, i.e. a solution to the equation $X\cong X\cup X$; we explore the many
consequences of this apparently innocuous observation. This has connection with
the category-theoretic analysis of dynamical systems.
