  We propose an integral representation for the higher-order Fr\'echet
derivative of analytic matrix functions $f(A)$ which unifies known results for
the first-order Fr\'echet derivative of general analytic matrix functions and
for higher-order Fr\'echet derivatives of $A^{-1}$. We highlight two
applications of this integral representation: On the one hand, it allows to
find the exact value of the level-2 condition number (i.e., the condition
number of the condition number) of $f(A)$ for a large class of functions $f$
when $A$ is Hermitian. On the other hand, it also allows to use numerical
quadrature methods to approximate higher-order Fr\'echet derivatives. We
demonstrate that in certain situations -- in particular when the derivative
order $k$ is moderate and the direction terms in the derivative have low-rank
structure -- the resulting algorithm can outperform established methods from
the literature by a large margin.
