This is a much easier way...I don't think you took your derivative right because the derivative of logx is not 1/x. It would be right if you change your logs to ln.
I don't see anything you can cancel out.
This seems a little bit misleading but maybe I am reading it poorly.
dy/dx(logx) is actually 1/x * 1/lnx, not 1/x * lnx which would put the lnx in the numerator instead of the denominator.