The derivatives of the given function $f(x) = x^4 + 3x^3 - 4x^{-1}$ are:

$f'(x) = 4x^3 + 9x^2 + \frac{4}{x^2}$

$f''(x) = 12x^2 + 18x - \frac{8}{x^3}$

$f'''(x) = 24x + 18 + \frac{24}{x^4}$

$f^{(4)}(x) = 24 - \frac{96}{x^5}$

Would you like any further details or have questions on these derivatives?

Here are five related questions to explore:

1. What are the critical points of the function $f(x)$?
2. How can the behavior of the function be analyzed using its second derivative?
3. What is the concavity of the function based on the second derivative?
4. How does the fourth derivative help in analyzing the function's higher-order behavior?
5. Can you find any inflection points for this function?

Tip: When calculating derivatives, always simplify expressions to avoid algebraic errors and make further analysis easier.