The main purpose of this paper is to study the hybrid mean value of $\frac {L'}L(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum _{\chi \neq \chi _0} |\tau (\chi )| |\frac {L'}L(1,\chi )|^{2k}$ of $\frac {L'}L$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.