how could you take the value of x here when there is no $x$ value in the derived equation, again i apologize if the question seems extremely simple this has legit got me stumped...
Basically i want to know how the $7$-th equation can have $x$ values when there are none left in the derived equation since the partial derivative is taken with respect to $x$.
I suspect you just misunderstood the statement:
They are not saying that $$\sum_n^H \sum_m^H U_{nm}^C \max(0, 1-|y_i^s-n|) = \begin{cases} 0 & \text{if } |m-x_i^s| \geq 1 \\ 1 & \text{if } m \geq x_i^s\\ -1 &\text{if } m < x_i^S\end{cases}.$$
Rather, they intend to say:
$$V_i^C = \sum_n^H \sum_m^H U_{nm}^C \max(0, 1-|x_i^s-m|) \max(0, 1-|y_i^s-n|)$$
$$\frac{\partial V_i^C}{\partial x_i^s} = \sum_n^H \sum_m^H U_{nm}^C \max(0, 1-|y_i^s-n|)\frac{\partial \max(0, 1-|x_i^s-m|)}{\partial x_i^S}$$
where
$$\frac{\partial \max(0, 1-|x_i^s-m|)}{\partial x_i^S} = \begin{cases} 0 & \text{if } |m-x_i^s| \geq 1 \\ 1 & \text{if } |m-x_i^s| < 1 \text{and } m \geq x_i^s\\ -1 &\text{if } |m-x_i^s| < 1 \text{and } m < x_i^S\end{cases}$$