I am trying to express the following idea: "studying the variation of function f1 and the variation of function f2 w.r.t. the number of active users N is essential for ....". Note that we can use these functions instead of f2 and f1. Here is my attempt:

Studying the variation of these rate functions w.r.t. the number of active users N is essential for ...

Is it correct ? can I express it better ?

I am trying to express the following idea: "studying the variation of function f1 and the variation of function f2 w.r.t. the number of active users N is essential for ....". Note that we can use these functions instead of f2 and f1. Here is my attempt:

Studying the variation of these rate functions w.r.t. the number of active users N is essential for ...

In this attempt, should I use variation or variations? is or are ?

I am trying to express the following idea: "studying the variation of function f1 and the variation of function f2 is/are essential for ....". Note that we can use these functions instead of f2 and f1. Here is my attempt:

Studying the variation of these rate functions w.r.t. the number of active users N is essential for ...

Is it correct ? can I express it better ?

I am trying to express the following idea: "studying the variation of function f1 and the variation of function f2 w.r.t. the number of active users N is essential for ....". Note that we can use these functions instead of f2 and f1. Here is my attempt:

Studying the variation of these rate functions w.r.t. the number of active users N is essential for ...

Is it correct ? can I express it better ?