  We establish a calculus of differences for taut endofunctors of the category
of sets, analogous to the classical calculus of finite differences for real
valued functions. We study how the difference operator interacts with limits
and colimits as categorical versions of the usual product and sum rules. The
first main result is a lax chain rule which has no counterpart for mere
functions. We also show that many important classes of functors (polynomials,
analytic functors, reduced powers, ...) are taut, and calculate explicit
formulas for their differences. Covariant Dirichlet series are introduced and
studied. The second main result is a Newton summation formula expressed as an
adjoint to the difference operator.
