The derivative of the function $y = x^4 + \frac{8}{x^2}$ with respect to $x$ is:

$\frac{dy}{dx} = 4x^3 - \frac{16}{x^3}$

Would you like more details on the steps involved in finding this derivative or have any questions?

Here are five related questions to expand on this topic:

1. What are the rules for differentiating a function involving powers of $x$?
2. How do we apply the power rule for negative exponents when differentiating?
3. What is the second derivative of the given function?
4. How can we find critical points using the first derivative?
5. How does the derivative behave for large values of $x$?

Tip: Always remember that differentiating terms with negative exponents follows the same power rule as positive exponents.