Possible derivation:
d/dx(d/dx((x^3-3 x y^2)/(x^2+y^2)^3))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 1/(x^2+y^2)^3 and v = x^3-3 x y^2:
  =  d/dx((x^3-3 x y^2) d/dx(1/(x^2+y^2)^3)+(d/dx(x^3-3 x y^2))/(x^2+y^2)^3)
Using the chain rule, d/dx(1/(x^2+y^2)^3) =  d/( du)1/u^3 ( du)/( dx), where u = x^2+y^2 and ( d)/( du)(1/u^3) = -3/u^4:
  =  d/dx((x^3-3 x y^2) (-3 d/dx(x^2+y^2))/(x^2+y^2)^4+(d/dx(x^3-3 x y^2))/(x^2+y^2)^3)
Differentiate the sum term by term:
  =  d/dx(((x^3-3 x y^2) -3 d/dx(x^2)+d/dx(y^2))/(x^2+y^2)^4+(d/dx(x^3-3 x y^2))/(x^2+y^2)^3)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
  =  d/dx(((x^3-3 x y^2) (-3 (2 x+d/dx(y^2))))/(x^2+y^2)^4+(d/dx(x^3-3 x y^2))/(x^2+y^2)^3)
The derivative of y^2 is zero:
  =  d/dx(((x^3-3 x y^2) (-3 (2 x+0)))/(x^2+y^2)^4+(d/dx(x^3-3 x y^2))/(x^2+y^2)^3)
Differentiate the sum term by term and factor out constants:
  =  d/dx(((x^3-3 x y^2) (-3 (2 x+0)))/(x^2+y^2)^4+d/dx(x^3)-3 y^2 d/dx(x)/(x^2+y^2)^3)
The derivative of x is 1:
  =  d/dx(((x^3-3 x y^2) (-3 (2 x+0)))/(x^2+y^2)^4+(1 -3 y^2+d/dx(x^3))/(x^2+y^2)^3)
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2:
  =  d/dx(((x^3-3 x y^2) (-3 (2 x+0)))/(x^2+y^2)^4+(-3 y^2+3 x^2)/(x^2+y^2)^3)
Differentiate the sum term by term and factor out constants:
  =  d/dx((3 x^2-3 y^2)/(x^2+y^2)^3)-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 3 x^2-3 y^2 and v = 1/(x^2+y^2)^3:
  =  -6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))+(d/dx(3 x^2-3 y^2))/(x^2+y^2)^3+(3 x^2-3 y^2) d/dx(1/(x^2+y^2)^3)
Simplify the expression:
  =  (d/dx(3 x^2-3 y^2))/(x^2+y^2)^3+(3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))
Differentiate the sum term by term and factor out constants:
  =  (3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))+3 d/dx(x^2)+d/dx(-3 y^2)/(x^2+y^2)^3
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
  =  (3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))+(d/dx(-3 y^2)+3 2 x)/(x^2+y^2)^3
Simplify the expression:
  =  (6 x+d/dx(-3 y^2))/(x^2+y^2)^3+(3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))
The derivative of -3 y^2 is zero:
  =  (3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))+(6 x+0)/(x^2+y^2)^3
Simplify the expression:
  =  (6 x)/(x^2+y^2)^3+(3 x^2-3 y^2) (d/dx(1/(x^2+y^2)^3))-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))
Using the chain rule, d/dx(1/(x^2+y^2)^3) =  d/( du)1/u^3 ( du)/( dx), where u = x^2+y^2 and ( d)/( du)(1/u^3) = -3/u^4:
  =  (6 x)/(x^2+y^2)^3-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))+(3 x^2-3 y^2) (-3 d/dx(x^2+y^2))/(x^2+y^2)^4
Differentiate the sum term by term:
  =  (6 x)/(x^2+y^2)^3-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))-(d/dx(x^2)+d/dx(y^2) 3 (3 x^2-3 y^2))/(x^2+y^2)^4
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
  =  (6 x)/(x^2+y^2)^3-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))-(3 (3 x^2-3 y^2) (d/dx(y^2)+2 x))/(x^2+y^2)^4
The derivative of y^2 is zero:
  =  (6 x)/(x^2+y^2)^3-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))-(3 (3 x^2-3 y^2) (2 x+0))/(x^2+y^2)^4
Simplify the expression:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 (d/dx((x (x^3-3 x y^2))/(x^2+y^2)^4))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x and v = (x^3-3 x y^2)/(x^2+y^2)^4:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2) d/dx(x))/(x^2+y^2)^4+x d/dx((x^3-3 x y^2)/(x^2+y^2)^4)
The derivative of x is 1:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 (x (d/dx((x^3-3 x y^2)/(x^2+y^2)^4))+(1 (x^3-3 x y^2))/(x^2+y^2)^4)
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 1/(x^2+y^2)^4 and v = x^3-3 x y^2:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+(x^3-3 x y^2) d/dx(1/(x^2+y^2)^4)+(d/dx(x^3-3 x y^2))/(x^2+y^2)^4 x)
Using the chain rule, d/dx(1/(x^2+y^2)^4) =  d/( du)1/u^4 ( du)/( dx), where u = x^2+y^2 and ( d)/( du)(1/u^4) = -4/u^5:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x ((d/dx(x^3-3 x y^2))/(x^2+y^2)^4+(x^3-3 x y^2) (-4 d/dx(x^2+y^2))/(x^2+y^2)^5))
Differentiate the sum term by term:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x ((d/dx(x^3-3 x y^2))/(x^2+y^2)^4-(d/dx(x^2)+d/dx(y^2) 4 (x^3-3 x y^2))/(x^2+y^2)^5))
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x ((d/dx(x^3-3 x y^2))/(x^2+y^2)^4-(4 (x^3-3 x y^2) (d/dx(y^2)+2 x))/(x^2+y^2)^5))
The derivative of y^2 is zero:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x ((d/dx(x^3-3 x y^2))/(x^2+y^2)^4-(4 (x^3-3 x y^2) (2 x+0))/(x^2+y^2)^5))
Simplify the expression:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x (-(8 x (x^3-3 x y^2))/(x^2+y^2)^5+(d/dx(x^3-3 x y^2))/(x^2+y^2)^4))
Differentiate the sum term by term and factor out constants:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x (-(8 x (x^3-3 x y^2))/(x^2+y^2)^5+d/dx(x^3)-3 y^2 d/dx(x)/(x^2+y^2)^4))
The derivative of x is 1:
  =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x (-(8 x (x^3-3 x y^2))/(x^2+y^2)^5+(d/dx(x^3)-1 3 y^2)/(x^2+y^2)^4))
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 3: d/dx(x^3) = 3 x^2:
Answer: |  
 |   =  -(6 x (3 x^2-3 y^2))/(x^2+y^2)^4+(6 x)/(x^2+y^2)^3-6 ((x^3-3 x y^2)/(x^2+y^2)^4+x (-(8 x (x^3-3 x y^2))/(x^2+y^2)^5+(-3 y^2+3 x^2)/(x^2+y^2)^4))