The problem asks for the derivative $f'(x)$ of the function $f(x) = \frac{1}{x^3}$.

Step-by-Step Solution:

1. Rewrite the function: $f(x) = \frac{1}{x^3} = x^{-3}$

2. Differentiate using the power rule: The power rule states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

   Applying this to $f(x) = x^{-3}$: $f'(x) = -3 \cdot x^{-4} = -\frac{3}{x^4}$

Conclusion:

The correct derivative is $f'(x) = -\frac{3}{x^4}$, which corresponds to the second option in the list.

Would you like more details, or do you have any questions?

Here are 8 related questions you might find helpful:

1. What is the power rule for differentiation, and how is it applied?
2. How do you differentiate functions that are in the form of $x^n$?
3. What are the common mistakes when differentiating functions like $f(x) = \frac{1}{x^n}$?
4. How do you differentiate functions that involve negative exponents?
5. Can you explain the chain rule and when it is used?
6. How do you find higher-order derivatives of functions like $f(x) = \frac{1}{x^3}$?
7. How does differentiation relate to the concept of a tangent line?
8. What are the practical applications of finding derivatives in real-world scenarios?

Tip: Always simplify the function, if possible, before differentiating to make the process easier.