f(x) = 3x\sqrt[3]{x^2} - 6x^{2/3}. Find f'(8).

We are given the function \( f(x) = 3x\sqrt[3]{x^2} - 6x^{2/3} \), and we are tasked with finding \( f'(8) \), the derivative of the function evaluated at \( x = 8 \).

### Step 1: Simplify the function
We can simplify the expression before differentiating.

- The term \( 3x\sqrt[3]{x^2} \) can be rewritten as \( 3x \cdot x^{2/3} = 3x^{1 + 2/3} = 3x^{5/3} \).
- So the function becomes:
  \[
  f(x) = 3x^{5/3} - 6x^{2/3}
  \]

### Step 2: Differentiate the function
To differentiate \( f(x) = 3x^{5/3} - 6x^{2/3} \), apply the power rule \( \frac{d}{dx} x^n = n x^{n-1} \).

1. For the first term, \( 3x^{5/3} \):
   \[
   \frac{d}{dx} \left( 3x^{5/3} \right) = 3 \cdot \frac{5}{3} x^{5/3 - 1} = 5x^{2/3}
   \]

2. For the second term, \( -6x^{2/3} \):
   \[
   \frac{d}{dx} \left( -6x^{2/3} \right) = -6 \cdot \frac{2}{3} x^{2/3 - 1} = -4x^{-1/3}
   \]

Thus, the derivative is:
\[
f'(x) = 5x^{2/3} - 4x^{-1/3}
\]

### Step 3: Evaluate the derivative at \( x = 8 \)
Now substitute \( x = 8 \) into the derivative:
\[
f'(8) = 5(8^{2/3}) - 4(8^{-1/3})
\]

We know that \( 8^{1/3} = 2 \), so:
\[
8^{2/3} = (8^{1/3})^2 = 2^2 = 4
\]
\[
8^{-1/3} = \frac{1}{8^{1/3}} = \frac{1}{2}
\]

Now substitute these values back into the expression for \( f'(8) \):
\[
f'(8) = 5(4) - 4\left(\frac{1}{2}\right) = 20 - 2 = 18
\]

### Final Answer:
\[
f'(8) = 18
\]

Would you like more details or clarifications on any part of the process?

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Here are 5 related questions to consider:

1. How would you find \( f'(x) \) for a function involving different types of radicals?
2. What is the significance of the power rule in calculus?
3. How does the chain rule apply to composite functions in differentiation?
4. Can you differentiate \( g(x) = x^{5/3} + x^{-1/3} \) and evaluate \( g'(8) \)?
5. How would you integrate the function \( f(x) = 3x^{5/3} - 6x^{2/3} \)?

**Tip**: When simplifying expressions with roots and powers, always look for opportunities to rewrite the expression in terms of exponents, which can make differentiation easier.

Find the derivative of the function f(x) = 3x^(5/3) - 6x^(2/3) and evaluate it at x = 8. This problem covers key calculus concepts such as the power rule and exponents, ideal for Grades 11-12 or early college-level calculus students.

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