f(x)=(ln x^2)^2 = (2lnx)^2 = 4 (lnx)^2

f'(x) = 4*2(lnx)*(1/x) = 4ln(x^2)/x

OR

Take it as a compound function, like:

g(u(x))=u(x)^2 where u(x)=ln x^2.

In that case:

f'(x) = 2u*u'(x) = 2u*(1/x^2)*(x^2)'=(2u/x^2)*2x

=4u/x

=4(lnx^2)/x

Either your answer key is wrong OR You are typing it wrongly here. OR I may be wrong

-O

4(ln x)^2/x