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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?

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**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.

Differentiate with respect to the independent variable: f(x) = 3x^3 - (5 / x^3)

Learn how to differentiate the function f(x) = 3x^3 - (5/x^3) with respect to x. This solution demonstrates the power rule and how to handle negative exponents, offering a detailed step-by-step explanation. Suitable for high school students in Grades 11-12.

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