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DETAILS SCALC9 2.5.002.
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Write the composite function in the form $f(g(x))$. [Identify the inner function $u=g(x)$ and the outer function $y=f(u)$.] (Use non-identity functions for $f(u)$ and $g(x)$.)
\[
y=\sqrt{x^{7}+6}
\]
$(f(u), g(x))=$
Find the derivative $\frac{d y}{d x}$.
\[
\frac{d y}{d x}=
\]
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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 ：\(\boxed{\frac{d y}{d x} = \frac{7x^{6}}{2\sqrt{x^{7}+6}}}\)