Motivated by the elaborate works of Forrest, Marcoux, and Zhang [Trans. Amer. Math. Soc. \\\\textbf{354} (2002), 1435--1452 and 4131--4151] on determining the first cohomology group and studying $n$-weak amenability of triangular and module extension Banach algebras, we investigate the same notions for a Morita context Banach algebra $\\\\K=\\\\left[\\\\begin{array}{cc} \\\\A & \\\\ \\\\M \\\\\\\\ \\\\N & \\\\ \\\\B% \\\\end{array}% \\\\right],$ where $\\\\A$ and $\\\\B$ are Banach algebras, $\\\\M$ and $\\\\N$ are Banach $(\\\\A,\\\\B)$ and $(\\\\B,\\\\A)$-bimodules, respectively. We describe the $n^{\\\\rm th}$-dual $\\\\K^{(n)}$ of $\\\\K$ and characterize the structure of derivations from $\\\\K$ to $\\\\K^{(n)}$ for studying the first cohomology group ${\\\\bf H}^1\\\\left(\\\\K, \\\\K^{(n)}\\\\right)$ and characterizing the $n$-weak amenability of $\\\\K$. Our study provides some improvements of certain known results on the triangular Banach algebras. The results are then applied to the full matrix Banach algebras $\\\\M_k(\\\\A)$. Some examples illustrating the results are also included, and several questions are also left undecided.

Keywords