Division in Banach Lattice Algebra

Question

Let X be a Banach Lattice Algebra and $X_+=\{f\in X: f>0\}$. Let $f:X_+\rightarrow X$ be continuously differentiable.

Question: When does the expression $\frac{f'(x)}{x}$ for $x\in X_+$, make sense? why is it allowed to divide by the positive element?
I saw such expression in an article, I want a justification for defining it.

Answer 1

Adapting reasonable axioms for a Banach lattice algebra, it follows that $X_+$ is open. As long as $X$ is commutative, this means $f^\prime(x) \cdot x^{-1}$ and it does make perfect sense since $f$ is differentiable.

Here positive most likely means strictly positive and such element should be invertible (again, it is not clear what is a Banach lattice algebra).