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)#]
