Suppose that $x$ and $y$ are related by the given equation and use implicit differentiation to determine $\frac{d y}{d x}$. \[ x^{7} y+y^{7} x=9 \] \[ \frac{d y}{d x}= \]

Step 1 ：Given the equation \(x^{7} y+y^{7} x=9\), we need to find \(\frac{d y}{d x}\).

Step 2 ：We can use the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 3 ：Applying the product rule to \(x^{7} y\), we get \(7x^{6} y + x^{7} \frac{d y}{d x}\).

Step 4 ：Applying the product rule to \(y^{7} x\), we get \(7y^{6} x + y^{7}\).

Step 5 ：So, the derivative of the given equation is \(7x^{6} y + x^{7} \frac{d y}{d x} + 7y^{6} x + y^{7} = 0\).

Step 6 ：We can rearrange this equation to solve for \(\frac{d y}{d x}\), which gives us \(\frac{d y}{d x} = -\frac{7x^{6} y + 7y^{6} x}{x^{7} + y^{7}}\).

Step 7 ：So, the derivative of \(y\) with respect to \(x\) is \(\boxed{-\frac{7x^{6} y + 7y^{6} x}{x^{7} + y^{7}}}\).