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Let's go through this step by step.

### 1. Derivative using the definition of derivative
The function given is:

\[
A(p) = 3p^3 + 5p
\]

Using the power rule for derivatives, we calculate:

\[
A'(p) = \frac{d}{dp}(3p^3) + \frac{d}{dp}(5p)
\]

- Derivative of \(3p^3\) is \(9p^2\) (using the power rule: \(\frac{d}{dp}(p^n) = np^{n-1}\))
- Derivative of \(5p\) is \(5\)

Thus, the derivative of the function is:

\[
A'(p) = 9p^2 + 5
\]

### 2. Domain of the function
The original function \(A(p) = 3p^3 + 5p\) is a polynomial. Polynomials are defined for all real values of \(p\). Therefore, the domain of the function is:

\[
\text{Domain of } A(p): (-\infty, \infty)
\]

### 3. Domain of the derivative
Similarly, the derivative \(A'(p) = 9p^2 + 5\) is also a polynomial, and it is defined for all real values of \(p\). Therefore, the domain of the derivative is:

\[
\text{Domain of } A'(p): (-\infty, \infty)
\]

### Summary
- \( A'(p) = 9p^2 + 5 \)
- Domain of \(A(p)\): \((- \infty, \infty)\)
- Domain of \(A'(p)\): \((- \infty, \infty)\)

Would you like any further explanation on these steps?

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Here are 5 follow-up questions related to this topic:
1. What is the second derivative of \(A(p)\)?
2. How do you find the critical points of \(A(p)\)?
3. What is the general form of a polynomial's derivative?
4. Can a polynomial have a restricted domain?
5. How do you find the tangent line to the graph of \(A(p)\) at a specific point?

**Tip:** For polynomial functions, the derivative is always a simpler polynomial, and they typically have the same domain.

Find the derivative of A(p) = 3p^3 + 5p and state the domain of both the function and its derivative.

Learn how to find the derivative of the polynomial function A(p) = 3p^3 + 5p using the power rule, and understand the domain of the function and its derivative. Suitable for Grades 11-12 calculus students.

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