Let $n\in\mathbb{N}_{\geq1}$ and $\phi\in C^{1}_{\text{c}}(\mathbb{R})$ be given and assume that $q\in BV(\mathbb{R})$. Then, we are wondering if the following finite difference approximation holds: $$ \lim_{\eta\rightarrow 0} \tfrac{n}{2\eta}\int_{\mathbb{R}}\big(q(x+\eta)-q(x-\eta)\big)\big(q(x)\big)^{n-1}\phi(x)\;\mathrm{d}x=-\int_{\mathbb{R}}\big(q(x)\big)^{n}\phi'(x)\;\mathrm{d} x. $$ For $n\in\{1,2\}$ this is true and can be shown by moving the shift in $\eta$ onto the test function $\phi$. Does this also hold for $n>2$?
Note that this question is a simplified version of this question where we assume that $q$ itself cannot change with respect to $\eta$.
It is false.
Choose any $b>a>0$. Define $q$ as
$$q(x)=\begin{cases}0,x\in(-\infty,0)\cup[2,\infty)\\a,x\in[0,1)\\b,x\in[1,2)\end{cases}$$
Take $\phi$ as any function in $C^1_c(\mathbb{R})$ that is $1$ on $[0,2]$ and $0$ outside $(-1,3)$.
Assuming the conjecture is true, we should have
$$\lim_{\eta\rightarrow 0}\frac{1}{\eta}\int_{\mathbb{R}}(q(x+\eta)-q(x-\eta))q(x)^{n-1}dx=0$$
However, the result is $a\cdot a^{n-1}+(b-a)\cdot(a^{n-1}+b^{n-1})-b\cdot b^{n-1}=a^{n-1}b-ab^{n-1}\neq 0$.