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The problem asks to:

1. Find the derivative of \( f(x) = 3 + 6x - 9x^2 \) using the definition of the derivative.
2. State the domain of the function.
3. State the domain of its derivative.

### Step 1: Finding the derivative

The function is \( f(x) = 3 + 6x - 9x^2 \).

To differentiate:
- The derivative of \( 3 \) is \( 0 \).
- The derivative of \( 6x \) is \( 6 \).
- The derivative of \( -9x^2 \) is \( -18x \).

Thus, the derivative \( f'(x) \) is:
\[
f'(x) = 6 - 18x
\]

### Step 2: Domain of the function
The function \( f(x) = 3 + 6x - 9x^2 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of the function is:
\[
(-\infty, \infty)
\]

### Step 3: Domain of the derivative
Similarly, the derivative \( f'(x) = 6 - 18x \) is also a polynomial and is defined for all real numbers. Therefore, the domain of the derivative is also:
\[
(-\infty, \infty)
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you apply the definition of the derivative using limits?
2. What is the second derivative of \( f(x) = 3 + 6x - 9x^2 \)?
3. How do you determine whether a function is increasing or decreasing using its derivative?
4. What is the geometric interpretation of the derivative of a function?
5. Can a polynomial have a restricted domain?

**Tip**: When dealing with polynomials, their domains are typically all real numbers unless they involve division by zero or square roots of negative numbers.

Find the derivative of f(x) = 3 + 6x - 9x^2 using the definition of the derivative. State the domain of the function and the domain of its derivative.

Learn how to find the derivative of f(x) = 3 + 6x - 9x^2 and determine the domain of both the function and its derivative. This problem covers differentiation of polynomials and is suitable for Grades 11-12.

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