${\underbrace{u_1\left(x,z_1\right)=v_1+{\dot{u}}_x}}_{Bydefinitionof{v}_1}={\overbrace{-\frac{\partial V_x}{\partial x}{g}_x\left(x\right)-k_1\left({\underbrace{z_1-u_x\left(x\right)}}_{e_1}\right)}}^{v_1}+{\overbrace{\frac{\partial u_x}{\partial x}\left({\underbrace{f_x\left(x\right)+g_x\left(x\right)z_1}}_{\dot{x}(i.e.,\frac{dx}{dt})}\right)}}^{{\dot{u}}_x(i.e.,\frac{d{u}_x}{dt})}$