Differentiate by parts: let u=x, du/dx=1; v=ln(2x), e^v=2x, e^vdv/dx=2, 2xdv/dx=2, dv/dx=1/x.
d(uv)/dx=vdu/dx+udv/dx=ln(2x)+x/x=ln(2x)+1.
Note that ln(2x)=ln(2)+ln(x) so differentiating ln(2x) gives the same result as differentiating x because ln(2) is a constant.