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The problem states:

**Let \( f(x) = x \sqrt[5]{x} \). Find \( f'(9) \).**

### Solution:

We are tasked with finding the derivative \( f'(x) \) of the function \( f(x) = x \sqrt[5]{x} \) and then evaluating it at \( x = 9 \).

1. **Rewrite the function in a simpler form:**

\[
f(x) = x \cdot x^{1/5} = x^{1 + 1/5} = x^{6/5}
\]

2. **Differentiate \( f(x) \):**

The derivative of \( x^n \) with respect to \( x \) is \( n x^{n-1} \), so:

\[
f'(x) = \frac{6}{5} x^{\frac{6}{5} - 1} = \frac{6}{5} x^{1/5}
\]

3. **Evaluate \( f'(x) \) at \( x = 9 \):**

\[
f'(9) = \frac{6}{5} \cdot 9^{1/5}
\]

Now, calculate \( 9^{1/5} \). This is the 5th root of 9. Using a calculator:

\[
9^{1/5} \approx 1.5518
\]

Thus,

\[
f'(9) \approx \frac{6}{5} \times 1.5518 \approx 1.8622
\]

So, \( f'(9) \approx 1.86 \).

Would you like more details or have any questions?

### Follow-up questions:
1. What is the general rule for differentiating power functions?
2. How would the derivative change if \( f(x) = x^n \) for any real number \( n \)?
3. What other rules of differentiation could apply to products like \( x \sqrt[5]{x} \)?
4. How does the chain rule apply when differentiating more complex radical expressions?
5. Can you explain how to approximate roots like \( 9^{1/5} \) without a calculator?

**Tip:** Always simplify expressions like \( x^a \cdot x^b \) by adding the exponents before differentiating!

Let f(x) = x * 5th root of x. Find f'(9).

Learn how to differentiate f(x) = x * 5th root of x and calculate f'(9). This solution walks through simplifying the function and using the power rule for differentiation. Suitable for advanced high school or college calculus students.

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