  The Fr\'echet derivative $L_f(A,E)$ of the matrix function $f(A)$ plays an
important role in many different applications, including condition number
estimation and network analysis. We present several different Krylov subspace
methods for computing low-rank approximations of $L_f(A,E)$ when the direction
term $E$ is of rank one (which can easily be extended to general low-rank). We
analyze the convergence of the resulting method for the important special case
that $A$ is Hermitian and $f$ is either the exponential, the logarithm or a
Stieltjes function. In a number of numerical tests, both including matrices
from benchmark collections and from real-world applications, we demonstrate and
compare the accuracy and efficiency of the proposed methods.
