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.