Partial difference operators for a large class of functors between presheaf categories are introduced, extending our previous work on the difference operator to the multivariable case. These combine into the Jacobian profunctor that provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered; however, we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the well-studied multivariable analytic functors.
