I am sorry, likely my initial question was wrong formulated.

Maybe this question would be more correct and easier:

How to prove

dm = m*v*dv / (c^2 - v^2)

is equivalent to

m = m0 / (1-(v^2 / c^2))^(1/2)

?

It is easy to write a program to calculate

dm = m*v*dv / (c^2 - v^2)

numerically by expression

m += m*v*dv / (c^2 - v^2)

inside a loop.

But I need analytical derivation.

Thank you