Step 1 :Identify the inner function $g(x)$ and the outer function $f(u)$ from the given function $y=\sqrt{x^{7}+6}$. The inner function $g(x)$ is $x^{7}+6$ and the outer function $f(u)$ is $\sqrt{u}$. So, $(f(u), g(x)) = (\sqrt{u}, x^{7}+6)$.
Step 2 :Find the derivative of the function using the chain rule. The derivative of the outer function $f(u)=\sqrt{u}$ is $\frac{1}{2\sqrt{u}}$, and the derivative of the inner function $g(x)=x^{7}+6$ is $7x^{6}$. Therefore, the derivative of the function $y=\sqrt{x^{7}+6}$ with respect to $x$ is $\frac{1}{2\sqrt{x^{7}+6}} \cdot 7x^{6}$.
Step 3 :Simplify the derivative to get the final answer. The derivative of the function $y=\sqrt{x^{7}+6}$ with respect to $x$ is $\frac{7x^{6}}{2\sqrt{x^{7}+6}}$. So, $\frac{d y}{d x} = \frac{7x^{6}}{2\sqrt{x^{7}+6}}$.
Step 4 :\(\boxed{\frac{d y}{d x} = \frac{7x^{6}}{2\sqrt{x^{7}+6}}}\)