Given the function \( f(x) = 3x^9 \), we can analyze and perform a variety of tasks such as differentiation, integration, and general analysis. Let me start by providing the derivative and integral of the function:

### 1. Derivative of \( f(x) \)
To differentiate \( f(x) = 3x^9 \) with respect to \( x \), apply the power rule:

\[
f'(x) = \frac{d}{dx}[3x^9] = 9 \cdot 3x^{9-1} = 27x^8
\]

So, the derivative of the function is:

\[
f'(x) = 27x^8
\]

### 2. Integral of \( f(x) \)
To find the indefinite integral of \( f(x) = 3x^9 \), apply the power rule of integration:

\[
\int 3x^9 \, dx = \frac{3x^{9+1}}{9+1} + C = \frac{3x^{10}}{10} + C = \frac{3}{10}x^{10} + C
\]

Thus, the indefinite integral is:

\[
\int f(x) \, dx = \frac{3}{10}x^{10} + C
\]

Do you have any specific questions or would you like further details on these calculations?

### Related Questions:
1. What is the second derivative of \( f(x) = 3x^9 \)?
2. How would you find the critical points of the function \( f(x) \)?
3. What is the definite integral of \( f(x) \) over a specific interval, say from \( x=0 \) to \( x=2 \)?
4. How does the graph of \( f(x) = 3x^9 \) behave as \( x \to \infty \) and \( x \to -\infty \)?
5. Can you solve for \( x \) when \( f(x) = 0 \)?

### Tip:
When differentiating or integrating powers of \( x \), always remember to apply the power rule carefully, adjusting the exponent accordingly.

Explore how to differentiate and integrate the polynomial function f(x) = 3x^9 with detailed solutions. Learn the power rule for differentiation and integration, ideal for students in Grades 11-12 or early university calculus courses.

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