Let \(\mathcal A\) be a triangular algebra. Let \(n\geq 2\) be an integer. A mapping \(\varphi\colon\mathcal A\times\mathcal A\times\cdots\times\mathcal A\to\mathcal A\) is said to be an \(n\)-derivation if it is a derivation in each argument. In this paper the authors mainly investigate \(n\)-derivations (\(n\geq 3\)) for a certain class of triangular algebras. Let \(\mathcal A=\text{Tri}(A,M,B)\) be a triangular algebra. Suppose that there exists \(m\in M\) such that \([m,[\mathcal A,\mathcal A]]=0\). Let us write \(\psi_n(x_1,x_2,\ldots,x_n)=[x_1,[x_2,\ldots,[x_n,m]\cdots]]\) for all \(x_1,x_2,\ldots,x_n\in\mathcal A\). Then \(\psi_n\) is a permuting \(n\)-derivation of \(\mathcal A\). One usually calls permuting \(n\)-derivations of the above form `extremal \(n\)-derivations'. An extremal \(2\)-derivation is said to be an `extremal biderivation'. Let \(\mathcal A=\text{Tri}(A,M,B)\) be a triangular algebra. If the following conditions are satisfied; (1) \(\pi_A(Z(\mathcal A))=Z(A)\) and \(\pi_B(Z(\mathcal A))=Z(B)\), (2) either \(A\) or \(B\) does not contain nonzero central ideals, (3) each derivation of \(\mathcal A\) is inner; then every \(n\)-derivation (\(n\geq 3\)) \(\varphi\colon\mathcal A\times\mathcal A\times\cdots\times\mathcal A\to\mathcal A\) is an extremal \(n\)-derivation. -- And then this main result is applied to upper triangular matrix algebras and nest algebras.