The following derivative can be solved by chain rule.

Let us substitute # (x-3)/x^2= u^2#

#:. sqrt ( (x-3)/x^2) = u#

Differentiating the original function #f# w.r.t. #u#,

#(df)/(du) = d/(du)(u)#

#=> (df)/(du) = 1 #

Now, we are required to determine #(df)/(dx)#

#(df)/(dx) = (df)/(du)*(du)/(dx)#

Differentiating w.r.t. #x# on both sides of the expression # (x-3)/x^2= u^2# we get ( applying quotient rule ),

#(x^2-(x-3)*2x)/x^4 = d/dx(u^2)#

#=>(6-x^2)/x^3 = (du)/dx.2u#

#=>(du)/dx = (6-x^2)/(x^3.(2u)#

#=>(du)/dx = (6-x^2)/(x^3.(2.sqrt ( (x-3)/x^2) )#

#:. (df)/dx = 1.(6-x^2)/(x^3.(2.sqrt ( (x-3)/x^2) )# [#(df)/(du)*(du)/(dx)#]