Given $x \in \mathbb{R}^N$ and a function $$H = \sum_{i,j,k=1}^n\ J_{i,j,k}\ x_i x_j x_k$$ for a fixed $J \in \mathbb{R}^{n \times n \times n}$, I am trying to calculate the derivative $\frac{d H}{d x}$ and a little lost in the expressions. Is there is a clean formula to do so? In the quadratic case, if $J$ is symmetric, the derivative is simply $J x$, is the expression simpler if $J_{i,j,k}$ is made symmetric, e.g., by adding all the 6 permutations together?
In general, if we have $$ H = \sum_{i_1,i_2 \ldots i_p=1}^n\ J_{i_1,i_2 \ldots i_p}\ x_{i_1} \ldots x_{i_p} $$ can we find a formula for the derivative?
Notice that $\dfrac{\partial x_p}{\partial x_q}=\delta_{pq}$.
Then
$$\dfrac{\partial H}{\partial x_m} = \sum_{i,j,k=1}^n\ J_{i,j,k}(\delta_{im} x_j x_k+x_i \delta_{jm} x_k+x_i x_j \delta_{km})=\\\sum_{j,k=1}^n\ J_{m,j,k} x_j x_k+\sum_{i,k=1}^n\ J_{i,m,k} x_i x_k+\sum_{i,j=1}^n\ J_{i,j,m} x_i x_j.$$