find the dy/dx of:

x^2=(x-y)/(x+y)

To find the derivative of a function with respect to x, we need to use the quotient rule when dealing with a fraction.

The equation you provided is: x^2 = (x - y)/(x + y)

Let's simplify it before finding the derivative:

1. Multiply both sides of the equation by (x + y) to eliminate the fraction:
(x + y) * x^2 = (x - y)

2. Expand the left side using the distributive property:
x^3 + xy = x - y

Now, we have our equation in a simplified form. To find dy/dx, we need to isolate y on one side of the equation and differentiate implicitly with respect to x.

3. Rearrange the equation to solve for y:
x^3 + xy = x - y
xy + y = x - x^3

4. Combine like terms:
y(x + 1) = x - x^3

5. Divide both sides by (x + 1) to solve for y:
y = (x - x^3)/(x + 1)

Now that we have an expression for y in terms of x, we can differentiate it implicitly to find dy/dx.

6. Apply the quotient rule:
dy/dx = [(x + 1)(1) - (x - x^3)(1)] / (x + 1)^2

7. Simplify the numerator:
dy/dx = (x + 1 - x + x^3) / (x + 1)^2

8. Combine like terms in the numerator:
dy/dx = (x^3 + 1) / (x + 1)^2

Therefore, the derivative dy/dx of the provided equation x^2 = (x - y)/(x + y) is (x^3 + 1) / (x + 1)^2.