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### Problem 1: Find the derivative of the function using the definition of derivative.

Given function:
\[
A(p) = 2p^3 + 9p
\]

To find the derivative \( A'(p) \), we use the power rule for differentiation:

1. **Differentiate** \( 2p^3 \):
   \[
   \frac{d}{dp}(2p^3) = 2 \cdot 3p^{3-1} = 6p^2
   \]

2. **Differentiate** \( 9p \):
   \[
   \frac{d}{dp}(9p) = 9
   \]

So, the derivative \( A'(p) \) is:
\[
A'(p) = 6p^2 + 9
\]

### Problem 2: State the domain of the function \( A(p) \).

The function \( A(p) = 2p^3 + 9p \) is a polynomial, and polynomials are defined for all real numbers. Thus, the domain of \( A(p) \) is:
\[
\text{Domain of } A(p): \ (-\infty, \infty)
\]

### Problem 3: State the domain of the derivative \( A'(p) \).

Similarly, the derivative \( A'(p) = 6p^2 + 9 \) is also a polynomial, and like all polynomials, it is defined for all real numbers. Therefore, the domain of \( A'(p) \) is:
\[
\text{Domain of } A'(p): \ (-\infty, \infty)
\]

### Summary of Answers:
1. \( A'(p) = 6p^2 + 9 \)
2. Domain of \( A(p) \): \( (-\infty, \infty) \)
3. Domain of \( A'(p) \): \( (-\infty, \infty) \)

Would you like further details or explanations on any part of this problem?

Here are 5 related questions to expand your understanding:
1. What is the definition of a derivative, and how does it apply to polynomial functions?
2. How would you find the second derivative of \( A(p) \)?
3. What is the geometric interpretation of the derivative \( A'(p) \)?
4. Could there be any cases where the domain of a derivative differs from the domain of the original function?
5. How does the degree of a polynomial affect the degree of its derivative?

**Tip:** Remember that the derivative of a polynomial of degree \( n \) will be a polynomial of degree \( n-1 \).

Find the derivative of A(p) = 2p^3 + 9p. Also, state the domain of the function and its derivative in interval notation.

Find the derivative of the function A(p) = 2p^3 + 9p using the power rule. The domain of the function and its derivative is also calculated. This problem covers differentiation of polynomials and domain analysis for students in Grades 11-12.

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