  We describe higher-order chain rules for multivariate functions and tensor
fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the
total variation seminorms of the higher derivatives of tensor fields after a
change of variables and determine sufficient regularity conditions for the
coordinate change. We also introduce a novel higher-order chain rule for
composition chains of multivariate functions that is described via nested set
partitions and generalized Bell polynomials; it is a natural extension of the
Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of
tensor calculus and includes Fr\'echet-differentiable mappings between Banach
spaces.
