Another approach to get derivative of odd-power

In this manuscript, we provide and discuss another approach to get derivative of odd-power such that is based on an identity in partial derivatives in terms of polynomial function $f_y$ defined as $$f_y(x,z)=\sum _{k=1}^z\sum _{r=0}^y{\mathbf{A}}_{y,r}{k}^r(x-k{)}^r$$ where $x,z\in \mathbb{R}$, $y$ is fixed constant $y\in \mathbb{N}$ and ${\mathbf{A}}_{y,r}$ are real coefficients.