Explicit closed-form real-variable expressions of a fundamental solution and its derivatives for three-dimensional problems in transversely linear elastic isotropic solids are presented. The expressions of the fundamental solution in displacements Uik and its derivatives, originated by a unit point force, are valid for any combination of material properties and for any orientation of the radius vector between the source and field points. An ex- pression of Uik in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector is used as starting point. Working from this expression of Uik, a new approach (based on the application of the rotational symmetry of the material) for deducing the first and second order derivative kernels, Uik,j and Uik,jl respectively, has been developed. The expressions of the fundamental solution and its derivatives do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational symmetry axis. The expressions of Uik, Uik,j and Uik,jl are presented in a form suitable for an efficient computational implementation in BEM codes.