I am trying to derive the following function with respect to $\Delta \tau^\alpha$, $\Delta g^\alpha$ and $\Delta \chi^\alpha$: $$ \dot{\gamma}_{t+\Delta t}^{\alpha} = \dot{a}sgn((\tau^\alpha+\Delta\tau^\alpha) - (\chi^\alpha+\Delta\chi^\alpha))\left |\frac{(\tau^\alpha+\Delta\tau^\alpha) - (\chi^\alpha+\Delta\chi^\alpha)}{g^\alpha + \Delta g^\alpha} \right |^n $$ I extracted the function from this paper: https://www.sciencedirect.com/science/article/pii/S0142112312001624 (page 215 and 14 in the pdf). The partial derivatives that I am trying to calculate are in the paper as follows: $$ \frac{\partial \dot{\gamma}_{t+\Delta t}^{\alpha}}{\partial \Delta \tau^\alpha} = \dot{a}n\left \{ \left | \frac{\tau^\alpha + \Delta \tau^\alpha - x^\alpha - \Delta x^\alpha}{g^\alpha + \Delta g^\alpha} \right | \right \}^{n-1}\frac{1}{g^\alpha+\Delta g^\alpha} $$ $$ \frac{\partial \dot{\gamma}_{t+\Delta t}^{\alpha}}{\partial \Delta g^\alpha} = -\dot{a}n\left \{ \left | \frac{\tau^\alpha + \Delta \tau^\alpha - x^\alpha - \Delta x^\alpha}{g^\alpha + \Delta g^\alpha} \right | \right \}^{n-1}\frac{\tau^\alpha + \Delta \tau^\alpha - x^\alpha - \Delta x^\alpha}{(g^\alpha+\Delta g^\alpha)^2} $$ $$ \frac{\partial \dot{\gamma}_{t+\Delta t}^{\alpha}}{\partial \Delta \chi^\alpha} = -\dot{a}n\left \{ \left | \frac{\tau^\alpha + \Delta \tau^\alpha - x^\alpha - \Delta x^\alpha}{g^\alpha + \Delta g^\alpha} \right | \right \}^{n-1}\frac{1}{g^\alpha+\Delta g^\alpha} $$ I am having doubts regarding the sgn function. Why it is not in the partial derivatives? Is that an error or am I missing something? Could you please explain to me how to obtain those expressions?
Thank you for your help.