We are tasked with finding the derivative of the function:

$f(x) = \sqrt{x} \left( 8x^8 + 6x^4 + 8x \right).$

To proceed, we will use the product rule for differentiation and also apply the chain rule where necessary.

Let me calculate the derivative for you.The derivative of the function $f(x) = \sqrt{x} \left( 8x^8 + 6x^4 + 8x \right)$ is:

$f'(x) = \sqrt{x} \left( 68x^7 + 27x^3 + 12 \right).$

Let me know if you have any further questions or need details!

Here are 5 related questions:

1. How would you apply the product rule in this case if $f(x)$ were more complex?
2. What are the steps for simplifying expressions involving square roots and polynomials?
3. How does the chain rule help when differentiating composite functions like this?
4. What are some examples of functions where you'd use the quotient rule instead of the product rule?
5. How does changing the exponents in the original polynomial affect the derivative?

Tip: Always look for common factors when simplifying expressions after differentiation—it can make the result much cleaner!