If I know the bound of the certain derivatives: $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_n}_n}V(x)|\leq M\epsilon^{-k} \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_n$ and $\epsilon$ is a small parameter. I also know that $$|\dfrac{\partial}{\partial x_j }V(x)|\leq M,$$ where $j\neq1$. How can I derive $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_{j-1}}_{j-1}\partial x_j\partial x^{k_{j+1}}_{j+1}\cdots \partial x^{k_n}_n}V(x)|\leq M(\epsilon^{-k_1}+\epsilon^{1-k}) \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_{j-1}+1+k_{j+1}+\cdots+k_n$? The function $g(x)$ is given but I think it does not matter in deriving the result. I think the problem is about differentiation rules.
I am trying to understand page 23 of this book