Euler's identity for the power potential $F_p$ and its gradient $G_p$

Let $p \in \mathbb R$ with $p>0$, and let $x \in \mathbb R$. Write $F_p(x) = |x|^p/p$ (powerPotential) and $G_p(x) = |x|^{p-2}x$ (powerGradient). Then

Equivalently, since $p\cdot F_p(x) = |x|^p$, the gradient times $x$ recovers $|x|^p$ exactly. This is the Euler identity for the potential $F_p$, which is positively homogeneous of degree $p$: multiplying its gradient by $x$ recovers $p$ times the potential itself. It is an algebraic bookkeeping fact used to simplify expressions that mix the power gradient and the power potential at the same point, arising when the Bregman divergence's defining formula is expanded and rearranged.

Formalization Note. The identity holds for every $p>0$ and every real $x$, including $x=0$ (both sides vanish) and the range $0<p\le1$, where $G_p(\cdot)$ is a totalized algebraic convention rather than an actual derivative of $F_p$ at the origin: $F_p$ itself is not differentiable at $0$ when $p\le1$.