To differentiate the function $f(x) = x^9$, we will use the power rule of differentiation. The power rule states:

$\frac{d}{dx} \left( x^n \right) = n x^{n-1}$

where $n$ is a constant.

Applying the power rule to $f(x) = x^9$:

$f'(x) = 9x^{9-1} = 9x^8$

So, the derivative of $f(x) = x^9$ is:

$\boxed{f'(x) = 9x^8}$

Would you like more details or have any questions?

Related Questions:

1. What is the derivative of $f(x) = x^{10}$?
2. How would you differentiate $f(x) = x^9 + x^8$?
3. What is the second derivative of $f(x) = x^9$?
4. Can you explain the power rule in more detail?
5. How do you differentiate functions that are products of different powers of $x$?

Tip: The power rule is a fundamental rule in calculus and is very useful for differentiating polynomial functions!