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?

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.