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)]