  Consider the set of unitary operators on a complex separable Hilbert space
$\mathcal{H}$, denoted as $\mathcal{U}(\mathcal{H})$. Consider $1<p<\infty$. We
establish that $f$ is $n$ times continuously Fr\'echet
$\mathcal{S}^{p}$-differentiable at every point in $\mathcal{U}(\mathcal{H})$
if and only if $f\in C^n(\mathbb{T})$. Take $U
:\mathbb{R}\rightarrow\mathcal{U}(\mathcal{H})$ such that
$\tilde{U}:t\in\mathbb{R}\mapsto U(t)-U(0)$ is $n$ times continuously
$\mathcal{S}^p$-differentiable on $\mathbb{R}$. Consequently, for $f\in
C^n(\mathbb{T})$, we prove that $f$ is $n$ times continuously G\^ateaux
$\mathcal{S}^p$-differentiable at $U(t)$. We provide explicit expressions for
both types of derivatives of $f$ in terms of multiple operator integrals. In
the domain of unitary operators, these results closely follow the $n$th order
successes for self-adjoint operators achieved by the second author, Le Merdy,
Skripka, and Sukochev. Furthermore, as for application, we derive a formula and
$\mathcal{S}^{p}$-estimates for operator Taylor remainders for a broader class
of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev
and Tomskova.
