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 (x). Express your answer in radical form without using negative exponents, simplifying all fractions.

To differentiate the given function \( f(x) = \frac{1}{3} \sqrt[3]{x^5} \), we will first express it in a more convenient form for differentiation.

### Step 1: Rewrite the function using exponent notation
The cube root of \( x^5 \) can be written as \( x^{\frac{5}{3}} \). Therefore, the function becomes:
\[
f(x) = \frac{1}{3} x^{\frac{5}{3}}
\]

### Step 2: Differentiate using the power rule
Now, we apply the power rule for derivatives, which states that if \( f(x) = ax^n \), then \( f'(x) = a n x^{n-1} \).

In this case, \( a = \frac{1}{3} \) and \( n = \frac{5}{3} \). Applying the power rule:
\[
f'(x) = \frac{1}{3} \times \frac{5}{3} x^{\frac{5}{3} - 1}
\]

### Step 3: Simplify the expression
First, calculate the new exponent:
\[
\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}
\]

Now, simplify the coefficient:
\[
f'(x) = \frac{5}{9} x^{\frac{2}{3}}
\]

### Step 4: Express the answer in radical form
Since \( x^{\frac{2}{3}} \) can be written as \( \sqrt[3]{x^2} \), the derivative in radical form is:
\[
f'(x) = \frac{5}{9} \sqrt[3]{x^2}
\]

### Final Answer:
\[
f'(x) = \frac{5}{9} \sqrt[3]{x^2}
\]

Do you want further details or have any questions?

Here are 5 related questions:
1. How do you differentiate a product of two functions?
2. What is the chain rule in differentiation, and how does it apply here?
3. How would the result change if the exponent was negative?
4. Can you explain how to handle more complex roots like fourth or fifth roots?
5. What happens if the function includes a logarithmic term?

**Tip**: When differentiating powers and roots, always rewrite them using exponents to make differentiation easier.

Given the function f(x) = (1/3) * cube root(x^5), find f'(x). Express your answer in radical form without using negative exponents, simplifying all fractions.

Find the derivative of the function f(x) = (1/3) * cube root(x^5) using the power rule of differentiation. This step-by-step guide explains how to rewrite radical expressions with exponents and simplify the result in radical form. Suitable for Grades 11-12 students studying calculus.

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