Two analytical procedures based on the time-dependent Hartree-Fock (TDHF) scheme are elaborated to evaluate the first derivatives of the dynamic first hyperpolarizability with respect to atomic Cartesian coordinates. In the first scheme, the mixed third-order TDHF equations are solved iteratively to obtain the third derivatives of the linear combination of atomic orbitals coefficients twice with respect to external dynamic electric fields and once with respect to atomic Cartesian coordinates. The second takes advantage of the 2n+1 rule and interchange relations to express the first derivatives of the dynamic first hyperpolarizability with respect to atomic Cartesian coordinates in terms of lower-order derivatives. Both procedures have been implemented in the GAMESS quantum chemistry package. Applications focus on the characterization of the frequency dispersion of the first derivatives of the first hyperpolarizability with respect to vibrational normal coordinates of H2O, NH3, and CH4 as well as on the determination of nonresonant hyper-Raman intensities within the double harmonic oscillator approximation.