  In this paper we consider the problem of the existence of higher derivatives
of the function $t\mapsto\f(A+tK)$, where $\f$ is a function on the real line,
$A$ is a self-adjoint operator, and $K$ is a bounded self-adjoint operator. We
improve earlier results by Sten'kin. In order to do this, we give a new
approach to multiple operator integrals. This approach improves the earlier
approach given by Sten'kin. We also consider a similar problem for unitary
operators.
