The first image derives macro to micro formula relation for T change for entropy. The other image derives micro to macro formula relation for V change.

I will try to explain my problem: I get both theese derivations. Isolated they both relate entropy quantifiably from macro to micro and vice versa. My problem is how I can quantify a path of both V and T change for entropy and make it as an explanation for spontanity mathematically? In other words how can one prove mathematically how much T entropy change and V entropy change matter relative to each other and relative to spontanity. I know that entropy is a state function and hence path independent.

I think I will try to explain my problem further with an example:

For a volume expansion dS=KlnW_f-klnW_i and after that for a temperature change: dS=KlnW_f-klnW_i. Now I don't know why they are like this relative to each other that is the volume entropy change and the temperature entropy change. I need an explanation for how strong the entropy force from volume change is relative to the force from temperature change since entropy governs the force spontanity. In order to get the spontanity force I need to know all the forces in entropy. And I need a marthematical explanation.