[x2 + y2]2 = 4x2y

Apply the chain rule to the left side and the product rule to the right side.

$$\frac{d}{dx}(x^2 + y^2)^2 = \frac{d}{dx}(4x^2y)$$ $$2(x^2 + y^2) \cdot \frac{d}{dx}(x^2 + y^2) = 4 \left( \frac{d}{dx}(x^2) \cdot y + x^2 \cdot \frac{d}{dx}(y) \right)$$ $$2(x^2 + y^2)(2x + 2y \frac{dy}{dx}) = 4(2xy + x^2 \frac{dy}{dx})$$

$$4(x^2 + y^2)(x + y \frac{dy}{dx}) = 8xy + 4x^2 \frac{dy}{dx}$$ $$4x(x^2 + y^2) + 4y(x^2 + y^2) \frac{dy}{dx} = 8xy + 4x^2 \frac{dy}{dx}$$ $$4x^3 + 4xy^2 + (4x^2y + 4y^3) \frac{dy}{dx} = 8xy + 4x^2 \frac{dy}{dx}$$

Move all terms containing $\frac{dy}{dx}$ to one side and other terms to the other side:

$$(4x^2y + 4y^3) \frac{dy}{dx} - 4x^2 \frac{dy}{dx} = 8xy - 4x^3 - 4xy^2$$

$$\frac{dy}{dx}(4x^2y + 4y^3 - 4x^2) = 8xy - 4x^3 - 4xy^2$$ $$\frac{dy}{dx} = \frac{8xy - 4x^3 - 4xy^2}{4x^2y + 4y^3 - 4x^2}$$

Divide the numerator and denominator by 4 to simplify:

$$\frac{dy}{dx} = \frac{2xy - x^3 - xy^2}{x^2y + y^3 - x^2}$$

The derivative $\frac{dy}{dx}$ is: