The problem is I do not know how to do the maths. Can someone give a clear step-by-step calculation on how that x value is derived?
If you have a little background in calculus, you'll notice that the dg(x)/dx line is the derivative of g(x) and then is set equal to 0 to find the critical point which in this case will be a maximum point and thus the optimal fraction of the bankroll to bet. Here is a little background/refresher on performing derivatives with log functions.
Whether or not you understand how to arrive at the derivative of log functions yourself, these particular ones boil down to a simple formula: the derivative of a*log(b+cx) = ac/(b+cx), where a, b, and c are constants.
With this example, you have three terms with their respective a, b, and c values being: (0.62, 1, -1); (0.24, 1, 1); (0.14, 1, 3). Plugging these into the above formula gets you to the dg(x)/dx line and from there you'll use some algebra to combine terms and solving for x.