In this paper, we introduce and solve the following additive-additive \((s,t)\) -functional inequality 1 \(\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \|g\left(x+y\right) -g(x) -g(y)\| +\| h(x+y) + h(x-y) -2 h(x) \| \\ & &\displaystyle \quad \le \left\|s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\| \\ & &\displaystyle \qquad + \left\|t \left( 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x)\right) \right\| , \end{array} \end{aligned} \) where s and t are fixed nonzero complex numbers with \(|s| <1\) and \( |t| <1\) . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of Lie bracket derivation-derivations in complex Banach algebras, associated to the additive-additive \((s,t)\) -functional inequality (1) and the following functional inequality 2 \(\displaystyle \begin{aligned} {}\| [g, h](xy)-[g,h](x) y- x [g,h](y) \| +\| h(xy) - h(x) y -x h(y) \| \le \varphi(x,y). \end{aligned} \)