Summary: Let \(A\) be a finite-dimensional, commutative algebra over \(\mathbb{R}\) or \(\mathbb{C} \). The notion of \(A\)-differentiable functions on \(A\) is extended to develop a theory of \(A\)-differentiable functions on finitely generated \(A\)-modules. Let \(U\) be an open, bounded and convex subset of such a module. An explicit formula is given for \(A\)-differentiable functions on \(U\) of prescribed class of differentiability in terms of real or complex differentiable functions, in the case when \(A\) is singly generated and the module is arbitrary and in the case when \(A\) is arbitrary and the module is free. Certain components of \(A\)-differentiable function are proved to have higher differentiability than the function itself. Let \(\mathsf{M}\) be a constant, square matrix. By using the formula mentioned above, a complete description of solutions of the equation \(\operatorname{grad}(w)=\mathsf{M}\operatorname{grad}(v)\) is given. A boundary value problem for generalized Laplace equations \(\mathsf{M}\nabla^2 v=\nabla^2v \mathsf{M}^{\mathsf{T} }\) is formulated and it is shown that for given boundary data there exists a unique solution, for which a formula is provided.