  Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:{\mathbb
R}\to {\mathbb C}$ is $n$-times Fr\'echet differentiable on ${\mathcal S}^p$ at
every self-adjoint operator if and only if $f$ is $n$-times differentiable,
$f',f'',\ldots,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.
