SOLUTION: if f ^(2 n)(x) + ((f (x) - 2))^(2 n) = x ^(2 n), n \[Element] N , then (dx)/(d (f (x))) at f (x) = 1 is ....

So, n is a natural number, i.e. positive integer number.


Differentiate the given equation.  You will get


    (2n)*f^(2n-1)(x)*f'(x) + (2n)*(f(x)-2)^(2n-1)*f'(x) = (2n)*x^(2n-1).


Cancel factor (2n) in both sides


    f^(2n-1)(x)*f'(x) + (f(x)-2)^(2n-1)*f'(x) = x^(2n-1).


Take  f'(x)  out the parentheses as a common factor 

   df
  ---- * [f^(2n-1)(x) + (f(x)-2)^(2n-1)] = x^(2n-1).
   dx


Then  

    dx     f^(2n-1)(x) + (f(x)-2)^(2n-1))
   ---- = ----------------------------------
    df                x^(2n-1)


Take it at  f(x) = 1


    dx     1^(2n-1) + (1-2)^(2n-1)      1 - 1
   ---- = -------------------------- = ----------.
    df             x^(2n-1)             x^(2n-1)


                         dx
So, if x =/= 0,  then   ---- = 0.   
                         df


But x definitely is not zero, since, otherwise, in the original equation left side would be 

     = 1 + 1 = 2,


while the right side would be   = 0.


                                     dx
Thus, the final conclusion is that  ---- = 0.    ANSWER
                                     df