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Let's work through the derivative of the given function:

The function is:

\[
f(x) = 9x^3 - \frac{1}{3x^2}
\]

To find the derivative \( f'(x) \), we will differentiate each term separately using basic differentiation rules.

### Step 1: Derivative of \( 9x^3 \)
- Use the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \).
- The derivative of \( 9x^3 \) is:
  \[
  \frac{d}{dx}(9x^3) = 9 \cdot 3x^{2} = 27x^2
  \]

### Step 2: Derivative of \( \frac{-1}{3x^2} \)
- First, rewrite \( \frac{1}{3x^2} \) as \( \frac{1}{3}x^{-2} \).
- Now apply the power rule to \( x^{-2} \):
  \[
  \frac{d}{dx}\left( \frac{1}{3}x^{-2} \right) = \frac{1}{3} \cdot (-2)x^{-3} = -\frac{2}{3}x^{-3}
  \]

### Step 3: Combine the results
Now, combining the two results:
\[
f'(x) = 27x^2 - \frac{2}{3}x^{-3}
\]

Therefore, the derivative of the function is:

\[
f'(x) = 27x^2 - \frac{2}{3x^3}
\]

Let me know if you'd like any further clarification or details.

### Questions to deepen understanding:
1. What is the power rule in differentiation, and how does it apply to polynomials?
2. How would you differentiate a term with a negative exponent like \( x^{-2} \)?
3. Can the quotient rule be applied instead of rewriting \( \frac{1}{3x^2} \)?
4. How does the chain rule differ from the power rule, and when would you use it?
5. What would happen if the function had an additional constant term, such as \( +5 \)?

### Tip:
When differentiating terms with fractions or negative exponents, it can be helpful to rewrite them in a simpler form using exponents before applying the power rule.

Find the derivative of the function: f(x) = 9x^3 - 1 / 3x^2

This problem demonstrates how to differentiate the function f(x) = 9x^3 - 1/(3x^2) using the power rule and negative exponents. A step-by-step solution is provided, suitable for high school students studying Calculus.

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