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!