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

Step 1 ：Given the equation $x^{7}y+y^{7}x=9$, we need to find $\frac{dy}{dx}$.

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 $7x^{6}y+x^7\frac{dy}{dx}$.

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 $7x^{6}y+x^7\frac{dy}{dx}+7y^{6}x+y^7=0$.

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

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