How to find dy/dx of the function y=x^(1/x) using logs and implicit differentiation?

Y = x ^(1/x)
Taking logs:-

ln y =  ln x^(1/x)
ln y  = 1/x  ln x

differentiating:-

dy/dx * 1/y =    = 1/x * 1/x + lnx * - 1/x^2

dy/dx =   (1/x^2 - lnx / x^2 ) * y

         =  [x ^(1/x) ( 1 - ln x) ]/ x^2  answer

I believe it would be A. because B is not true, you would have enough information to make a call. C is incorrect because the range in the fourth grade book is only 6, while the range of the eighth grade book is 9. and D is false because even if you take the mean plus the range to find the longest book, the eight grade book would still have a longer word