  The tensor t-function, a formalism that generalizes the well-known concept of
matrix functions to third-order tensors, is introduced in [K. Lund, The tensor
t-function: a definition for functions of third-order tensors, Numer. Linear
Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the
Fr\'echet derivative of the tensor t-function and derive algorithms for its
efficient numerical computation. Applications in condition number estimation
and nuclear norm minimization are explored. Numerical experiments implemented
by the \texttt{t-Frechet} toolbox hosted at
\url{https://gitlab.com/katlund/t-frechet} illustrate properties of the
t-function Fr\'echet derivative, as well as the efficiency and accuracy of the
proposed algorithms.
