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.