I want to derive the 4th order Runge-Kutta method. I want to know how to get the coefficient in very detail.
I've read all book about it, possible duplicate, many journals, sites but all of them not telling anything about choosing this coefficient. How do I get $1, \frac12, \frac16, \frac18$, etc? How do I get the constant in the quadrature and sub-quadrature equations?
I did not say it was obvious to me. I was saying it would be obvious after you worked out the derivation of the lower RK's. Start with the Euler method, and justify why you would want to choose coefficients in a certain way. As the previous commenter mentioned, and provided papers for, it often has to do with the order of approximation the method requires, in this case, 4.