When should I use "by" in present participle?

I often encounter a problem when writing sentences in mathematical contexts, where one "does something" in order to "obtain something." These sentences typically have the form "using X, one can do Y," where generally X is a mathematical rule or function, and Y is its result. I never know if I should put "by" before the present participle. I have this problem mainly with the verb "using," but this is not the only verb that gives me trouble.

Here are some examples:

(by) using Theorem 2, one can prove that...

We obtain the result (by) expanding the function ...

(by) differentiating w.r.t. the 2nd variable, we obtain ...

(by) taking the integer part of F, one can compute...

Comments

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