Find $f^{\prime \prime}(x)$
\[
f(x)=\left(x^{2}+8\right)^{5}
\]
\[
f^{\prime \prime}(x)=
\]

Step 1 ：Given the function \(f(x)=(x^{2}+8)^{5}\), we are asked to find the second derivative \(f''(x)\).

Step 2 ：We start by finding the first derivative \(f'(x)\) using the chain rule. The chain rule states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function.

Step 3 ：Here, our outer function is \(u^5\) and our inner function is \(x^2+8\). So, \(f'(x) = 5u^4 \cdot (2x) = 10x(x^2 + 8)^4\).

Step 4 ：Next, we find the second derivative \(f''(x)\) by applying the chain rule again. This time, our outer function is \(10xu^4\) and our inner function is still \(x^2+8\).

Step 5 ：So, \(f''(x) = 40xu^3 \cdot (2x) + 10u^4 = 80x^2(x^2 + 8)^3 + 10(x^2 + 8)^4\).

Step 6 ：\(\boxed{f''(x) = 80x^2(x^2 + 8)^3 + 10(x^2 + 8)^4}\) is the second derivative of the function \(f(x)=(x^{2}+8)^{5}\).