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We establish the following results on higher order$\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator functionarising from a continuous scalar function $f$ and self-adjoint operatorsdefined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuouslyFr\'{e}chet $\mathcal{S}^p$-differentiable at every bounded self-adjointoperator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\inC_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ timescontinuously Fr\'{e}chet $\mathcal{S}^p$-differentiable at every self-adjointoperator; (iii) if $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then $f$ is $n-1$times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable and $n$ timesG\^{a}teaux $\mathcal{S}^p$-differentiable at every self-adjoint operator. Wealso prove that if $f\in B_{\infty1}^n(\mathbb{R})\capB_{\infty1}^1(\mathbb{R})$, then $f$ is $n$ times continuously Fr\'{e}chet$\mathcal{S}^q$-differentiable, $1\le q<\infty$, at every self-adjointoperator. These results generalize and extend analogous results of [10] toarbitrary $n$ and unbounded operators as well as substantially extend theresults of [2,4,19] on higher order $\mathcal{S}^p$-differentiability of $f$ ina certain Wiener class, G\^{a}teaux $\mathcal{S}^2$-differentiability of $f\inC^n(\mathbb{R})$ with $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, and G\^{a}teaux$\mathcal{S}^q$-differentiability of $f$ in the intersection of the Besovclasses $B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$. As anapplication, we extend $\mathcal{S}^p$-estimates for operator Taylor remaindersto a broad set of symbols. Finally, we establish explicit formulas forFr\'{e}chet differentials and G\^{a}teaux derivatives.

Higher order differentiability of operator functions in Schatten norms