We establish, for 1<p<\infty , higher order \mathcal{S}^{p} -differentiability results of the function \varphi \colon t\in \mathbb{R} \mapsto f(A+tK) - f(A) for selfadjoint operators A and K on a separable Hilbert space \mathcal{H} with K element of the Schatten class \mathcal{S}^{p}(\mathcal{H}) and f n -times differentiable on \mathbb{R} . We prove that if either A and f^{(n)} are bounded, or f^{(i)} , 1\leq i\leq n , are bounded, \varphi is n -times differentiable on \mathbb{R} in the \mathcal{S}^{p} -norm with bounded n th derivative. If f\in C^n(\mathbb{R}) with bounded f^{(n)} , we prove that \varphi is n -times continuously differentiable on \mathbb{R} . We give explicit formulas for the derivatives of \varphi , in terms of multiple operator integrals. As for application, we establish a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a more extensive class of functions. These results are the n th order analogue of results by Kissin–Potapov–Shulman–Sukochev. They also extend the results of Le Merdy–Skripka from n -times continuously differentiable functions to n -times differentiable functions f .