y=F(x)\\ \end{array} \right .$$

The above result can be associated with the derivative of an $s-$plateaued function. The derivative of a vectorial function is defined by $$D_aF(x)=F(x+a)-F(x).$$

To see the connection let us first replace $y$ in the expression $$y=F(s+x-a)-F(s)+F(a)$$ by $F(x)-c$ for $c \in \mathbb{F}_{p^n}$. Hence we have $$c=F(s)-F(a)-F(s+x-a)+F(x)=D_{s-a}F(a)-D_{s-a}F(x).$$ This observation yields $$\begin{aligned}
 &\sum\limits_{a \in \mathbb{F}_{p^{n}}}|\{s \in \mathbb{F}_{p^{n}}\colon  y=F(s+x-a)-F(s)+F(a)\}|\\
 =&\sum\limits_{a \in \mathbb{F}_{p^{n}}}|\{s \in \mathbb{F}_{p^{n}}\colon  D_{s-a}F(a)-D_{s-a}F(x)=c\}|\\
 =&\sum\limits_{a \in \mathbb{F}_{p^{n}}}|\{t \in \mathbb{F}_{p^{n}}\colon  D_tF(a)-D_tF(x)=c\}|\\
 =&|\{(t,a) \in \mathbb{F}_{p^{n}}\times \mathbb{F}_{p^{n}}\colon  D_tF(a)-D_tF(x)=c\}|\\
 =&N_F(c, x)
 \end{aligned}$$ where $N_F(c, x)$ represents the number of pairs $(t,a) \in \mathbb{F}_{p^{n}}\times \mathbb{F}_{p^{n}}$ such that $$D_tF(a)-D_tF(x)=c$$ as in Section 2 of [@mos1]. Thus we will have the following result concerning the derivative and PGDS parameters.

\[diff2-plateaued\] Let $F$ be a function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p^m}$. Then the set $G_F$ is a PGDS with parameters $(p^{n+m},p^n ;\alpha, \beta)$ if and only if $$N_F(c, x)=  \begin{cases} 
    \alpha, & c\ne 0 \\
    \beta, & c = 0 \\
    \end{cases}$$ for all $x \in \mathbb{F}_{p^n}$ and some constants $\alpha$ and $\beta$.

Using Theorem \[diff2-plateaued\], Proposition \[diff-plateauedprop\] and Theorem \[plateaued-PGDS\], we are able to prove Theorem 8i. in [@mos1] with a different approach using the properties of partial geometric difference sets. This gives an interesting relation 