Willing to pay fg for someone to completely derive this

These sites do a decent job explaining it:

http://math.stackexchange.com/questions/528856/explanation-and-proof-of-the-fourth-order-runge-kutta-method

https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Derivation_of_the_Runge.E2.80.93Kutta_fourth-order_method

http://www.ijcaonline.org/volume9/number8/pxc3871891.pdf

I personally enjoyed reading the third.