To differentiate the function $f(x) = 3x^3 - \frac{5}{x^3}$ with respect to $x$, we will handle each term separately.

Step-by-Step Solution

1. Rewrite the function to simplify the differentiation process: $f(x) = 3x^3 - 5x^{-3}$

2. Differentiate each term with respect to $x$:

   • For $3x^3$: Use the power rule, $\frac{d}{dx}(x^n) = nx^{n-1}$. $\frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2$

   • For $-5x^{-3}$: Again, apply the power rule. $\frac{d}{dx}(-5x^{-3}) = -5 \cdot (-3)x^{-3-1} = 15x^{-4}$

3. Combine the results: $f'(x) = 9x^2 + 15x^{-4}$

4. Rewrite if necessary (optional): $f'(x) = 9x^2 + \frac{15}{x^4}$

Final Answer

$f'(x) = 9x^2 + \frac{15}{x^4}$

Would you like more details on any specific step, or do you have further questions?

Related Questions:

1. How do you apply the power rule in differentiation?
2. What are the steps to simplify a function before differentiating?
3. How do you handle negative exponents when differentiating?
4. What is the significance of rewriting a function before differentiating?
5. How can the power rule be applied to terms with fractional exponents?

Tip: When differentiating terms with negative exponents, remember to keep track of the negative signs as they affect the result.