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?

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.