Given the equation \( \sqrt{x^{2}+4 x y+y^{6}}=8 \), evaluate \( \frac{d y}{d x} \). Assume that the equation implicitly defines \( y \) as a differentiable function of \( x \). If \( F(x, y)=\sqrt{x^{2}+4 x y+y^{6}}-8=0 \), then \( F_{x}=\frac{x+2 y}{\sqrt{x^{2}+4 x y+y^{6}}} \) If \( F(x, y)=\sqrt{x^{2}+4 x y+y^{6}}-8=0 \), then \( F_{y}=\frac{2 x+3 y^{5}}{\sqrt{x^{2}+4 x y+y^{6}}} \) \( \frac{d y}{d x}= \)

To evaluate \( \frac{dy}{dx} \), we can use implicit differentiation on the given function \( F(x, y) = \sqrt{x^2 + 4xy + y^6} - 8 = 0 \). First, we know that: \[ \frac{dF}{dx} = F_x + F_y \frac{dy}{dx} = 0 \] From your provided information, we have: \[ F_x = \frac{x + 2y}{\sqrt{x^2 + 4xy + y^6}} \quad \text{and} \quad F_y = \frac{2x + 3y^5}{\sqrt{x^2 + 4xy + y^6}} \] Plugging these into the differentiation equation: \[ \frac{x + 2y}{\sqrt{x^2 + 4xy + y^6}} + \frac{2x + 3y^5}{\sqrt{x^2 + 4xy + y^6}} \frac{dy}{dx} = 0 \] Now, isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = -\frac{F_x}{F_y} = -\frac{\frac{x + 2y}{\sqrt{x^2 + 4xy + y^6}}}{\frac{2x + 3y^5}{\sqrt{x^2 + 4xy + y^6}}} = -\frac{x + 2y}{2x + 3y^5} \] Thus the final result is: \[ \frac{dy}{dx} = -\frac{x + 2y}{2x + 3y^5} \]