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