thanking you, but the first line is just xy(x^2-y^2), not (x^2 y^2 (x^2-y^2))
d/dx((x y(x) (x^2-y(x)^2))/(x^2+y(x)^2))
| Use the quotient rule, d/dx(u/v) = (( du)/( dx))/v-(u ( dv)/( dx))/v^2, where u = x y(x) (x^2-y(x)^2) and v = x^2+y(x)^2:
= | (d/dx(x y(x) (x^2-y(x)^2)))/(x^2+y(x)^2)-(x y(x) (x^2-y(x)^2) (d/dx(x^2+y(x)^2)))/(x^2+y(x)^2)^2
| Differentiate the sum term by term:
= | (d/dx(x y(x) (x^2-y(x)^2)))/(x^2+y(x)^2)-(x y(x) (x^2-y(x)^2) (d/dx(x^2)+d/dx(y(x)^2)))/(x^2+y(x)^2)^2
| The derivative of x^2 is 2 x:
= | -(x y(x) (x^2-y(x)^2) (d/dx(y(x)^2)))/(x^2+y(x)^2)^2+(d/dx(x y(x) (x^2-y(x)^2)))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| Use the chain rule, d/dx(y(x)^2) = ( du^2)/( du) ( du)/( dx), where u = y(x) and ( du^2)/( du) = 2 u:
= | -(2 x y(x)^2 (x^2-y(x)^2) (d/dx(y(x))))/(x^2+y(x)^2)^2+(d/dx(x y(x) (x^2-y(x)^2)))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| The derivative of y(x) is y'(x):
= | (d/dx(x y(x) (x^2-y(x)^2)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x and v = y(x) (x^2-y(x)^2):
= | (x (d/dx(y(x) (x^2-y(x)^2)))+y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = y(x) and v = x^2-y(x)^2:
= | (x (y(x) (d/dx(x^2-y(x)^2))+(x^2-y(x)^2) (d/dx(y(x)))))/(x^2+y(x)^2)+(y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| Differentiate the sum term by term and factor out constants:
= | (x y(x) (d/dx(x^2)-d/dx(y(x)^2)))/(x^2+y(x)^2)+(x (x^2-y(x)^2) (d/dx(y(x))))/(x^2+y(x)^2)+(y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| The derivative of x^2 is 2 x:
= | -(x y(x) (d/dx(y(x)^2)))/(x^2+y(x)^2)+(x (x^2-y(x)^2) (d/dx(y(x))))/(x^2+y(x)^2)+(y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2+(2 x^2 y(x))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| Use the chain rule, d/dx(y(x)^2) = ( du^2)/( du) ( du)/( dx), where u = y(x) and ( du^2)/( du) = 2 u:
= | -(2 x y(x)^2 (d/dx(y(x))))/(x^2+y(x)^2)+(x (x^2-y(x)^2) (d/dx(y(x))))/(x^2+y(x)^2)+(y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2+(2 x^2 y(x))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| The derivative of y(x) is y'(x):
= | (x (x^2-y(x)^2) (d/dx(y(x))))/(x^2+y(x)^2)+(y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 y'(x))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2+(2 x^2 y(x))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| The derivative of y(x) is y'(x):
= | (y(x) (x^2-y(x)^2) (d/dx(x)))/(x^2+y(x)^2)-(2 x y(x)^2 y'(x))/(x^2+y(x)^2)+(x (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2+(2 x^2 y(x))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2
| The derivative of x is 1:
= | -(2 x y(x)^2 y'(x))/(x^2+y(x)^2)+(x (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)-(2 x y(x)^2 (x^2-y(x)^2) y'(x))/(x^2+y(x)^2)^2+(2 x^2 y(x))/(x^2+y(x)^2)-(2 x^2 y(x) (x^2-y(x)^2))/(x^2+y(x)^2)^2+(y(x) (x^2-y(x)^2))/(x^2+y(x)^2)